Like you Dan, I too have a generalized aversion to anime. The whole "I pull sledgehammers out of my butt", "if you yell you become thirty feet tall", and "giant sweat drops mean embarrassment" are stylistic things I just can't get onboard with.
Beyond that, one of my friends and I were watching Dragon Ball Z way back when, and Freeza shoots a giant ball of energy into the planet Namek and goes, "Oh, it didn't immediately blow up. Looks like it has five minutes until it explodes." Five thirty-minute episodes later, Namek explodes. It was then that we stopped watching DBZ forever (except for abridged).
All that being said, anime I do enjoy:
Some of Cowboy Bebop (sorry, have to). Toys in the Attic ranks as one of the best episodes of any show ever. But, then again, there are plenty of episodes that are just irrelevant fluff.
Sword Art Online. Nearly every episode has its own self-contained story arc that contributes to the larger season/half-season arcs, and the English dubbing isn't half bad (when they get around to it). This is not to say that some episodes aren't slow (some of them are), just that they're self-contained, and a slow episode here and there is often a good thing for a series (the fly episode in Breaking Bad, for instance).
Trigun. I should be clear, I haven't watched Trigun since I was, like, 12, so it may not have aged well. But it was good when I was 12. You may want to start with the movie then move onto the series if you liked the movie.
Hellsing. Oh my, Hellsing. Hellsing is probably my favorite anime. It's difficult to pull of a rich, dark world that's not all broody and tortured, but I think they managed to pull it off (even if the main girl is broody and tortured some of the time) just because Alucard has so much fun. Also, Walter the Angel of Death.
Anime that you should watch because culture:
Akira. It's just one movie, you can do it. If you want to understand post-WWII Japan a little better, as well as the cyberpunk genre, you have to watch Akira. Warning: it gets... weird.
Ghost in the Shell. It's... hard to get through, and even harder to understand. But, if you want to understand where a lot of modern cyberpunk is coming from (apparently, it's always cyberpunk), you have to get through Ghost in the Shell.
I think I'm going to go watch Hellsing again. Oooh, it's on Hulu.