I think the full moon excuse is complete and utter rubbish. Factor 12 (possibly 13) full moons a year, multiplied by 26 years for me, and that's a lot of full moons. However, I have yet to wrap my car around a telephone pole, set fire to a neighbor's house, kidnap and bury another living creature, or even lose my temper unjustly during lunar cycles.
In most scenarios, it's a psychological thing. Since we were young we've been programmed to notice the difference between a full moon and a new moon, waxing and waning crescents. And there's always stories related to it... Werewolves, witches, hell even ET had a spot of fun with a full moon.
It's just like Friday the 13th, as mentioned somewhere else. I think people tend to pay more attention to the insignificant days or lunar phases than they do any other day, and that's when people begin to notice the "unusual" things.
As for the criminal cases, how many more officers were on duty during the full moon because of a psychological effect? I guarantee you that if you keep staffing consistent, the "Full Moon Effect" will be null and void.