One way to get particles that move faster than light without violating special relativity, which is what the OPERA experiment seemed to measure in neutrinos produced at CERN is with tachyons, particles that are moving backward in time.
The discussion has focused on superluminal neutrinos. But, even if something is moving at a rate faster than light, it wouldn't necessarily have to be the neutrinos. A proton-proton collision is a chain reaction. One of the component parts of the proton emits a W boson that in turn decays to something and a neutrino headed in the direction of the OPERA experiment.
It could be that the W boson is the tachyon (after all, the W boson does all sorts of other weird things that no other Standard Model particle does like CP violation, flavor changing, charge changing, having mass while being a boson rather than the fermion, etc.), while the neutrino itself is just a minding its own business, utterly boring, Lorentz invariant, Standard Model particle. Since high energy Ws are profoundly more rare and have much shorter lifetimes in nature than high energy neutrinos, this would lead to fewer observable effects.
The W would have to be be massively tachyonic to get a 60*10^-9 second effect with an observed 3*10-25 second half life. But, once one is in the realm of the weird, why not go whole hog. After all, the estimated half life of a W boson is based on the assumption that it is not tachyonic, and the considerations that went into estimating the W boson half life or direct measurements of its speed, have not been subject to nearly so much critical scrutiny as the OPERA experiment results. If a high energy W boson leaps backward in time 60 nanoseconds or so before emitting a neutrino and whatever else it is decaying into in this experiment, then nothing else in nature would ever have to act superluminal. And, the highly random nature of W production and decay might even make it impossible to send superluminal signals, even though particles created as a result of W boson decay might appear to move superluminally without more careful analysis of other alternatives. This might ease concerns about causality violations via tachyons.
Perhaps the reason that a W would be superluminal would be that bosonic masses are properly treated as negative masses in the theory of special relativity, while fermionic masses are properly treated as positive masses in the theory of special relativity. There are, after all, many respects in which bosons and fermions behave differently, and W and Z bosons are so short lived that their properties can only be measured indirectly. And, since W boson energy isn't connected to the energy of a neutrino produced in a W boson decay in a particularly straightforward manner, any energy-effect relationship between W boson energy might be obscured in a measurement looking for a connection between the effect observed and neutrino energy.
The possibility that neutrinos do indeed behave differently due to an imaginary Majorana mass component, meanwhile, is explored at the Quantum Diaries Survivor blog, in a discussion of this paper.