* Older ancient DNA both from the LBK area and era in Germany, where G2 and F* are found, and from Treilles, France (ca. 3000 BCE), where G2 is found as are subtypes of haplogroup E still found in Europe. Otzi the Ice Man from about 3000 BCE, is also G2a in Y-DNA haplogroup. All of these early European Neolithic examples of ancient Y-DNA lack R1a and R1b.
* The oldest purported R1b ancient DNA finds in Europe of which I am aware are from a 670 CE grave in Bavaria.
* R1b was also purportedly found in King Tut and his grandfather (whose earliest definitive paternal ancestors date to the 16th to 17th centuries depending on how intent one is upon being definitive). But, the King Tut information is partially rumor and the rumor is not specific enough to distinguish African centered R1b-V88 (particularly common among Chadic language speakers of the African Sahel) from the R1b haplogroup types found in Europe. King Tut's patriline is specific to Upper Egypt on the Sudanese side of the country, rather than Lower Egypt (i.e. Northern Egypt). His paternal ancestors ruled an Uper Egyptian kingdom at a time when the Semitic language speaking Hyskos prevailed in Lower Egypt in one of the Intermediary periods.
The data are obviously thin. But, the little data we do have suggests that G2a and perhaps also some haplogroups of E and F* were important in the early Neolithic, both LBK and Cardial Pottery. Y-DNA haplogroups R1a and R1b are completely absent in both Neolithic ancient Y-DNA datasets.
While direct evidence concerning paleolithic Y-DNA is almost absent, there is a considerable amount of paleolithic mtDNA available (almost all U4 and U5) and this shows a stark contrast with the mtDNA of the Neolithic and later eras. It is hard to imagine a scenario in which Europe experiences dramatic mtDNA shifts, while very little Y-DNA shifts consistent with the archaeological cultures we know to have existed then. History has shown major Y-DNA shifts without nearly as great mtDNA shifts much more often than the reverse, and the available physical anthropology also suggests a major Paleolithic/Neolithic shift in the LBK area, at least.
The ancient Y-DNA data also suggest that R1a was dominant in the Indo-European societies of Central and Eastern Europe and Central Asia in the Bronze Age (and in some places, earlier dating back to the Copper Age aka Eneolithic period), suggesting a major demographic shift between and perhaps as a result of, the succession from early Neolithic to Indo-European in these areas (at least, that is, in areas where there was food production in the first place).
Perhaps this shift took place with the with Corded Ware culture, or perhaps it took place with an earlier culture, and the demographic shift wouldn't necessarily have had to coincide with the cultural shift in all places. But, there are few good candidates between the LBK and Corded Ware to bring about such a decisive shift that are not themselves in strong continuity with Corded Ware.
The data are not inconsistent with the Bell Beaker civilization as the primary source of R1b in Western Europe, but the data also don't show the origins or timing of R1b's appearance on a widespread basis in Western Europe with direct evidence.
The distribution patterns one would expect from an expansion from a Franco-Cantabrian refuge in the Epipaleolithic are not so different from what one would expect from an Eneolithic Bell Beaker expansion, for example, since the source points would be roughly the same relative to non-Iberian Europe, and since the means of transportation weren't necessarily profoundly different. Similarly, one could imagine R1a being associated with one of the Epipaleolithic refugia from which Europe was repopulated that was centered further East, rather than from Indo-Europeans, based on geographic distribution alone.
The R1a/R1b boundary in Europe, while muddy at the boundaries (reflecting perhaps, the shifting extent of the Corded Ware and Bell Beaker cultures respectively), is pretty close to the Western limit of Corded Ware. The cultures belonging to the Corded Ware culture and their Western neighbors were apparently engaged in stalemated military conflict with each other at about that boundary for something on the order of a thousand years.
One could imagine the LBK and Cardial Pottery branches of the early European Neolithic (or perhaps two branches of the LBK Neolithic) squaring off against each other at the R1a/R1b boundary, but would have to explain why both sides were Y-DNA haplogroup G2 dominated in that scenario. Of course, the genetics look the same if Bell Beaker is an Indo-European source of R1b in Europe as it does if Bell Beaker is a non-Indo-European source of R1b in Europe, and the identity between the anti-Corded Ware front in the West with Bell Beaker isn't as perfectly established as one might hope. There is suggestive evidence to point towards the anti-Corded Ware fron as non-Indo-European, and a deep linguistic/cultural clash despite the use of similar technologies would certainly help to explain why the two sides would stay at odds with each other for centuries, but that evidence is pretty thin.
The Corded Ware and Bell Beaker cultures were roughly "contemporaneous with the Sumer period in Mesopotamia, the Early Dynastic period in Egypt and the early Troy (I and II)."
The Minoans ca. 3500 BCE-1450 BCE were also contemporaneous with them. The Minoans had a script that probably recounted a language rather than a proto-language in a somewhat phonetic way, but it has not been deciphered and the extant Minoan Linear A script appears to consist mostly of bureaucratic records of ration programs, taxes and religious sacrifices. My own impression is that Minoan bore a striking similarity to the non-Indo-European language of the Hattic people whose language was superceded by the Indo-European Hittites in the time period from ca. 2000 BCE to 1500 BCE, by which time Hattic had ceased to be used outside religious ceremonies.
Unlike some of the their more Southerly contemporaneous cultures, the Corded Ware and Bell Beaker cultures were apparently illiterate (or at best proto-literate) and were located in places largely outside the awareness of the literate cultures of their era. The first literate accounts of Indo-Europeans are from Egyptian and Sumerian records from ca. 2000 BCE and thereafter and describe the Hittites and Mycenean Greeks, who are from the areas closest to the literate world.
Any time one thinks about non-Indo-European cultures in Europe, another question that hops out at you is the larger linguistic affiliations and origins of the Etruscans, one of the last well attested non-Indo-European language of Europe to die (Raetic in the Alps survived to the 3rd century CE). The Etruscans came into their own as a culture in what is now Northern Italy (and in particular Tuscany), but appear to have been intrusive iron age arrivals there from the Alps, who had many cultural and technological similarities with contemporaneous, Indo-European Italic and Urnfield cultures. Arguably, they survived so long because they borrowed so much from the adversaries and hence were able to compete with them. The Roman Empire ultimately crushed them around 0 CE. Sometimes the Etuscans are associated with the other non-Indo-European languages of the Aegean (e.g. Minoan), but given their lack of obvious maritime origins, this hypothesis isn't necessarily solid.
The ancient mtDNA evidence do agree with a view of the Etruscans as intrusive, or at least, divergent from the rest of Italy:
Genetic distances and sequence comparisons show closer evolutionary relationships with the eastern Mediterranean shores for the Etruscans than for modern Italian populations. All mitochondrial lineages observed among the Etruscans appear typically European or West Asian, but only a few haplotypes were found to have an exact match in a modern mitochondrial database, raising new questions about the Etruscans’ fate after their assimilation into the Roman state.
They are an outlier European compared to modern populations as viewed in a principle component analysis chart.
On one dimension the Etruscans are at an extreme in the same direction as the Azerbaijanis, with the Italian Ladins, Druze, Egyptians, Near Easterners and Central Asians at the other extremes, and the bulk of Europeans in a cluster closer to the Etrustcans than to the outliers in the other direction with Danes, Armenians and Basques at one side of the cluster and Estonians, Sardians and Turks at the more distant side of that cluster.
On the other dimension the Estruscans are at an extreme in the same direction as the Italian Ladins and with Haviks of India, Central Asians, and Spanish Catalans as outliers in the other direction, and Sicilians, Southern Italians, Turks and Armenians at the distant side of the core European cluster, while the Swiss, Austrians, Danes and Icelanders are at the close side of the cluster.
Modern Tuscany is rich in Y-DNA haplogroup G. Tuscan cattle are closer genetically to cattle from the Near East than that of other parts of Italy or Europe. Other genetic studies link Tuscans with at least some Turks.
One narrative connects the Etruscans to the Lydians of Southwestern Anatolia, in accordinance with Herodotus writing in the 5th century BCE. But, there are good linguistic reasons to doubt that analysis because ca. 1200 BCE, when the Etruscan culture began to emerge, non-Indo-European languages had been pretty much squeezed out of existence in Anatolia between the Indo-European speaking Hittites and related Anatolian Indo-European languages that emerged when the Hittite empire fell in Bronze Age collapse around 1200 BCE (the Lydian language, for example, was an Anatolian Indo-European language), and the Afro-Asiatic language family members to the South.
There are arguments to connect Etruscan to Northwest Caucuasian, a connection that could arguably be traced to the Cardial Pottery Neolithic or an exodus of non-Indo-European Anatolians (e.g. Hurrians) to an Alpine refugium at some point. Notably, one of the proposed places of origin for the Cardium Pottery Neolithic is a close match to the place Herodotus attributes their origin to at a time when Indo-European languages were probably not spoken there, and largely skips over the Aegean.
(Genetically, the case to link them in some way to the Bell Beaker people via the Stele people seems very weak if those people were the main source of R1b in Western Europe and perhaps an important source of European cattle as well, although linguistically, the Stele people would make a good source for Raetian-Etruscan).
The timing of early Villanovan (i.e. proto-Etruscan) culture emergence suggests that some sort of push factor or vacuum created by Bronze Age collapse drove a migration from the Raetian Alps to Northern Italy. Pliny writing in the 1st century who proposes that the Raetians were driven out of their homeland into the mountains by the Celtic Gauls and are of "the same race" as the Etruscans seems closer to the mark.
Overall, the notion of Etruscans as legacies of the early Neolithic who survived by copying technologies and cultural traits from invading Indo-European Italo-Celtic peoples seems like one of the stronger interpretations available. The similarity to Northwest Caucasian, Hurrian languages probably reflects the fact that there are the best attested examples of languages related to the language most prominent in the early European Neolithic (of which Hurrian rather than Northwest Caucasian may very well be the more basal form).