There's a lot of parts of anime that piss me off...
I really, really hate that whole kawaii thing. The innocent girls that act like babies, the protagonist tries to help her, then falls over backwards and she screams at him with extremely annoying animation.
The battle scenes that take several episodes. Dragonball Z was awesome as a kid, but now I can't justify spending 3 hours watching two guys smack-talk over 6 episodes before they throw a punch or two.
The Final-Fantasyesque Fem-Men. I really dislike the androgynization of men in Japanese/asian culture. Not only is it evident in these media, but also in real people. There's so many fem-men on campus here. And girls fucking love it. Ugh.
Other that, though, I wouldn't mind watching a few. I used to watch the obvious, Pokemon, Monster Rancher, Digimon, and DBZ as a kid. I watched a few episodes of Cromartie High School last year, but stopped because of time. I also think I'd quite like Ghost in the Shell.
The anime style is very cool if done in a tasteful manner, like the Kill Bill anime sequence or some of the Animatrix shorts. However, in the whole, I think I can live without it and that I'd rather read the manga of the animes that are based on manga. (but then there's the time issue again, which means I've only read Death Note)