I'm... fairly selective, surprisingly.
Yeah, I watched K-On, and I just called myself selective.
Put your submachine guns down. Let me explain...
A show has to meet one major requirement for me to like it:
Is it something that is more objectionable than a typical American war movie?
That's my standard, because, let's face it, anime does typically go overboard, and I'm a Catholic (No, not in name only. I just tend to be really irreverent. But that's another story for another day).
Violence is fine (And loved!) so long as it's not pools of blood without reason. Cute lolis are alright, but giant breasted women tend to grate on the nerves very quickly.
Other than that, an anime has to be able to hold my (fairly limited) attention, and motivate me to move on to the next episode.
Shows like Geass and Azumanga succeeded at this (Although, hilariously enough, Azumanga probably wouldn't pass my test now. I've grown sick of comedy anime), while shows like R.O.D. tend to fail, just because the episodes were extremely... boring and generic. Read or Die OVA was awesome, though.
Additionally, a show has to not contain mass amounts of squick. Because this is more subjective, I really don't use it that heavily. But... I tend to like anime that doesn't make me shiver with the thought of extreme imagined pain for the next few hours.
Because of this, I have not seen Higurashi yet.