The predominant Y-DNA haplogroup of the first farmers of Europe was Y-DNA haplogroup G2, together with some Y-DNA haplogroup I, the predominant Y-DNA haplogroup of European hunter-gathers (probably integrated into the farmer community from hunter-gatherers), and I believe, a bit of Y-DNA halogroup T. This is consistent across the LBK farmers who were the first farmers of Central and Eastern Europe, the Cardial Pottery farmers who were the first farmers of Southern Europe, and the early Neolithic megalithic farmers of Western Europe.
Ancient DNA evidence strongly suggests that these first farmers replaced or demographically overwhelmed and assimilated in small percentages, the pre-existing hunter-gather men of Europe.
Relict populations, rich in Y-DNA haplogroup G, all of which likely date to this first wave of farmers, are found in several populations in the modern Caucasus mountains and Sardinia. Y-DNA haplogroup G is also found at elevated levels in Corsica, Gascony (in Southern France), and Tuscany (which was home to the Etuscans of Italy before they suffered ethnocide within the Roman Empire).
Y-DNA haplogroup G has its greatest diversity in the Levant, which was part of the Fertile Crescent where farming was invented, suggesting that it originated there. But, it is now most common in the Caucasus and the vicinity to the north of these mountains, where it is far less diverse.
Ancient DNA informs us that Y-DNA haplogroups R1a and R1b were almost entirely absent from the first farmers of Europe, appearing only in the Copper Age when they became the predominant Y-DNA haplogroups of Europe, diluting Y-DNA haplogroup G (and the small percentages of Y-DNA haplogroups I and T found in association with it) to low percentage frequencies in most of Europe.
Y-DNA R1b-V88, found mostly in Chadic people in Africa, arrived around 5700 BCE in the vicinity of Lake Chad, show signs of a linguistic connection to and wife taking from Cushitic people, probably in Southern Sudan. But, Y-DNA R1b-V88 was not shared by any other African population to any great extent.
Y-DNA haplogroups J1 and J2 were also associated with early Neolithic peoples, although just how this played out is a bit unclear. Today, J1 is confined largely to Semitic peoples who were traditionally herders in SW Asia. Meanwhile J2 is associated with the highlands of the Fertile Crescent - Anatolia, the Caucasus, Armenia, and the Zargos Mountains, none of whom have ever been linguistically Semitic in attested history. There is a cline of relative J2 to J1 proportions from these highlands where J2 is more common, to the Bedouins of the Arabian desert, where J1 is predominant, but in most of SW Asia, there is a mix of both types in any given population. Y-DNA haplogroup J is rare in North Africa and East Africa (and almost absent elsewhere in Africa). Where it is found, Y-DNA J1 predominates. Y-DNA J1 in East Africa is probably attributable mostly to Ethio-Semitic conquest ca. 1000 BCE, and in North Africa is probably mostly attributable to Islamic expansion and intercourse after 630 CE. The genetic diversity of Y-DNA J is similar throughout Europe, West Asia and SW Asia, leaving a precise point of origin unclear.
The nearly complete absence of Y-DNA haplogroup J in the ancient DNA of the first farmers of Europe suggest that like Y-DNA R1a and R1b, that its presence outside SW Asia is due to a later migration wave. It is also the case that outside SW Europe, Y-DNA J is predominantly Y-DNA J2, Y-DNA J1 in Europe is largely confined to Jews who migrated there in the historic era, and to areas where there was Muslim rule in the historic era. The expansion of Y-DNA J2 into Europe may have been as a minor component of the initially Y-DNA R1a dominated expansion of the Indo-Europeans into Europe in second or subsequent waves of farmer and herder migration into Europe.
Europe was not the only receiver of the Fertile Crescent Neolithic package of farming, herding and other technologies and cultural traits. The Fertile Crescent Neolithic package also spread to the Indus River Valley via diffusion at roughly the same rates seen in Europe across Iran, and to Egypt and beyond in North Africa.
Beyond these areas, the Fertile Crescent Neolithic package hit ecological and geographic barriers. Fertile Crescent crops did not thrive in the climate of the African Sahel or Sub-Saharan Africa, and domesticated Fertile Crescent animals could not survive the tropical diseases of tropical Africa. Fertile Crescent crops likewise were not suitable for the climate of tropical, monsoon driven Southern India. The European and Central Asian steppe could support the herding of Fertile Crescent domesticated animals, but was too dry for the farm crops in the Fertile Crescent Neolithic package of the first farmers to thrive without inventing arid region optimized irrigation technologies (which eventually were invented).
The secondary centers of the Neolithic revolution that ultimately produced the Harappan civilization of the Indus River Valley, and the secondary center in Egyptian and North Africa, however, follow a very different demographic pattern than the demographic pattern observed in Europe.
In the Indus River Valley, prior to the arrival of Indo-Aryan invaders around 2000 BCE, Y-DNA haplogroups L and R2, as well as autochronous South Asian Y-DNA haplogroups like Y-DNA haplogroup H were common, while Y-DNA haplogroups R1, J2 and G that are predominant in early European and Near Eastern farmers were largely absent until the Indo-Aryans arrived in the wake of the collapse of Harappan society in the face of a severe climate event. Y-DNA T is found in India, but largely in Dravidian areas of Southeastern India that did not adopt agriculture until around 2500 BCE using mostly African Sahel crops. Y-DNA halpogroup T probably arrived in India at that time, and is otherwise largely absent from South Asia.
We don't know if Y-DNA L and R2 arrived in the Indus River Valley at the same time, or if they represent separate waves of migration to the region. We do know that Y-DNA L is the sister clade to Y-DNA T and that the two clades may have geographically close origins to each other. And, we do know that there are strong indications that the split of Y-DNA R into subclades R1 and R2 and possibly also into R1a and R1b, took place in Iran, which has maximal Y-DNA diversity today. All Y-DNA R appears to have remote origins in the early Upper Paleolithic era in Southeast Asia and a very basal form of Y-DNA R was found in the ancient DNA of Ma'alta man from ca. 24,000 years ago in the Altai Mountain region at the far Southeast of the Eurasian steppe.
It is reasonable to hypothesize that the first farmers were Europe were from the Levant at the west end of the Fertile Crescent, while the first farmers of the Indus River Valley and Iran were from ethnically and genetically distinct populations probably originating from Mesopotamia and the Zargos Mountains at the east end of the Fertile Crescent (an area that in the Copper Age had thriving maritime trade with the Indus River Valley).
We also know that the Fertile Crescent Neolithic package involved several independent domestication events at different centers of population within the Fertile Crescent, that were then assembled into a comprehensive package, probably through trade and exchanges of small numbers of farming experts. It was not invented by a single anthropological culture and the Fertile Crescent was rarely under the control of a single state or ruler in historic times. So, it makes sense that different populations that developed particular components of the Fertile Crescent Neolithic package might have different population genetics.
In Egypt, which received the Neolithic revolution from the Fertile Crescent at about the same time as the Indus River Valley and the Balkans in Europe, there are traces of Y-DNA haplogroups G and T, but there is far more Y-DNA haplogroup T than there is Y-DNA haplogroup G. But, there is very little Y-DNA haplogroup R1 in Egypt that is not traceable to the historic era. And the Y-DNA haplogroup E clades, typical of linguistically Afro-Asiatic areas that probably pre-date the Neolithic revolution are very common. Thus, in Egypt, rather than the population replacement we see in Europe, we see a much milder demographic impact. (King Tut was probably Y-DNA R1b, but that was likely attributable to the historic area Semitic Hyskos invasion and establishment of their Egyptian dynasties, or to Greek contacts with Egypt, not to the deep population genetic history of Egypt.)
This isn't to say that the Neolithic revolution didn't bring about immense demographic change in Egypt and North Africa. Population densities increased by as much as a hundred-fold in a matter of a few centuries. Any population that did not participate in the demographic population expansion that North Africa experienced as a result of the Neolithic revolution would be almost invisible in modern North African population genetics.
But, unlike Europe, the Neolithic Revolution in Egypt and North Africa was not largely a story of pure male population replacement by outside populations. There is good evidence from the distribution and phylogeny of Y-DNA haplogroup E's subclades that it originated in the vicinity of Ethiopia, not in SW Asia as a back migrating clade.
Perhaps the fishing economy, and abundance of the hunting and gathering in the Nile Valley gave Egypt staying power in the face of Levantine Neolithic migrants in a way that was not true in Europe. And, Egypt was the gateway through which all Neolithic expansion in Africa was filtered.
It is also possible, for example, that Y-DNA G and Y-DNA T may have represented two distinct populations in the Levant. Perhaps Y-DNA G was made up of farmers and Y-DNA T was made up of herders. In Europe, which was well suited to farming, Y-DNA G dominated, while Y-DNA T left a thin shadow of its range with much less demographic impact.
In contrast, in Egypt and beyond in North Africa and East Africa, herding was a much more significant component of the Neolithic revolution as there wasn't nearly as much arable land available to farm upon due to the narrowness of the Nile River Valley. So, the impact of the Y-DNA T herder migrants from the Fertile Crescent may have been comparatively great and the impact of the Y-DNA G migrants from the Fertile Crescent may have been comparatively small.
Furthermore, historical evidence seems to suggest that it is much easier for people to transition from a terrestrial hunter-gather mode of subsistence to a herder mode of subsistence, than it is to transition from a terrestrial hunter-gather mode of subsistence to farming. So, cultural transmission of the Neolithic revolution may have been easier in North Africa where that mostly involved teaching people how to be herders, than it was in Europe where that involved teaching people how to be farmers. Hence, there may have been less economic pressure for the people who were the source of the Neolithic revolution to replace the existing population of North Africa than there was to replace the existing population in Europe (and probably also in Iran and Indus River Valley).
On the other hand, that specific scenario doesn't square very well with the fact that the Y-DNA T clades found in Southern Arabia are much younger than those in the Levant, or the fact that in SW Asia, low land herding is largely conducted by men who are Y-DNA J1 rather than Y-DNA T.
In general, the timing and circumstances by which Y-DNA J entered into its large demographic role in SW Asia, especially Y-DNA J1, is not clear.
UPDATED December 18, 2014 to correct numerous spelling, punctuation, grammar, and usage issues, as well as a few inadvertent errors, and to better spell out some incomplete thoughts. Every once and a while when I'm in a really pinch for time, I come up with a real stinker in terms of formal writing errors like these. I've also looked at Maju's comment and done a bit a research of Y-DNA J2 since this post, but I am not ready to see if I can add some substance to the discussion of Y-DNA J in this post.