These days, I don't watch too many anime.
I start with anime that is relevant to my tastes. Anime that is inspired by toku or based on things I all ready loved. That's why I am a fan of Sonic X and Powerpuff Girls Z. I also liked animes like Karas and Busou Renkin, but Busou is better in its Japanese dub (When the English Voice Actors don't know how to emote or REALLY drive their lines in, then it can be annoying).
I also try to keep up with that's popular and relevant, but half the time those kind of animes annoy me to high heaven. I grew up on Dragonball, so seeing how far it has come in the states never gets tiring to hear about. I use to watch Sailor Moon, and I still think Funimation should grab the rights and release it uncut (Maybe then we can also get the live action show in the states). And sometimes other shows, like Code Geass, One Piece, or a number of Gundam series will capture my attention. But shows like Naruto and Bleach just too annoying for my tastes. I guess it all depends on the writing and the mix the stories have.
EDIT: Forgot to mention that there are also things that can drive me away from anime besides bad or repetitive story telling. The over usage of blood and gore, the over usage of things like ecchi and panty shots and maybe even nudity, massive and senseless murders, a complete lack of story, or a story too confusing and technical to follow. I kind of like my animes to be followable, to mean something, and to be fun. I also am not a fan of animes that tell stories from the perspective of the the villains. I am a hero supporter through and through! I am sure there are other factors I am forgetting about, but I think that should be enough for now.