DAMA's claimed signal is seen in flashes of scintillation light (not Cerenkov, sorry Cuddles) in 100kg of ultrapure NaI. The flashes seen are in the range of 2-6 keV, far below the range where Cerenkov light occurs. They see about 1 flash per kg-day-keV in their detector constantly, but they claim a ~0.01 flash/kg/d/kev higher rate in the summer and ~0.01 flash/kg/d/kev lower rate in the winter. It is certain that most of these flashes are backgrounds from radioactive decays in the detectors, the shielding, etc.., but such decays should not vary from season to season.
The *expected* dark matter signal is when a Galactic dark matter particle crashes into a nucleus. The incoming dark matter velocity is the sum of several components: (a) some random isotropic velocity left over from the formation of the Galaxy, (b) 220 km/s because the Sun is moving through the galaxy, and (c) 10 km/s because the Earth is moving around the Sun. You'd expect the dark matter signal to be higher in the summer when the Earth's motion *adds* to the Sun's (with respect to the Galactic rest frame) and lower in the winter when the motions partially cancel. The size of this effect is expected to be in the ballpark of a few percent.
The problem is, if DAMA is seeing 0.01 c/d/kg/kev of *variation* between 4 and 6 keV ... why did XENON10, a much cleaner experiment, see fewer than 0.003 c/d/kev/kg *total events* between 4 and 6 keV? (XENON10 was not sensitive to the 2-4 keV events which DAMA saw.)
If DAMA's signal to be dark matter, it is very very difficult to imagine (a) why no other experiments have seen it and (b) why the annual modulation would be so large a fraction of the signal. If this is the case, the dark matter involved must be something very weird---axions, with weird galactic dynamics, coupling to electrons? MeV-scale WIMPs coupling to nuclear spin? Something else?
DAMA is expected to see some neutrino events, yes, but (I think) nowhere near enough to explain their large signals---remember that most neutrino detectors are kilotonnes, not 100 kilograms. It's hard to think of a neutrino source that makes such low-energy events, and it's even harder to think of a neutrino source which is 10% stronger in summer than in winter!
The thing to remember is that DAMA's signal has nothing to do with simply seeing events. Everyone sees events: XENON, CDMS, COUPP, etc.. What's new about DAMA is this claim of seasonal variation.
The other thing to remember is that seasonal variation is *really hard* to get right. Cosmic-ray fluxes vary over the year. Temperatures vary, which might couple to radon infiltration rates, noise, etc. If could be something like, "The incompetent students from University X do the calibrations during their summer vacation, while expert postdocs from University Y are available all winter." It's a very difficult measurement, and an easy one to get subtly wrong. That said, everything I could think of has been given a careful treatment in their paper.