You don't really sound a like a big anime guy, TC, and like me probably can't get into the REALLY Japanese animes. Granted, I like a lighter tone in my work than you seem to, but I'll go ahead and give my thoughts.
I agree with Samurai Champloo. I consider it every bit the equal of Bebop, only with a break-dancing samurai.
I DO NOT recommend Evangelion. Maybe you'll like it, but I fucking hated it. It's already depressing from the get go but then it goes down an infinite well of disgusting and emo and never comes back up. I've never wanted to punch a cartoon character as much as I have Shinji, and I've never wanted to admit a person to a psych ward as much as that show's creator.
Death Note I thought was pretty good. Even after the "halfway point" where many people say it sucked, I thought it did a decent job. Not as good as the beginning, but I think that was the point. The whole point of the second half of the series was to show that the protagonist is NOT SOMEONE TO ROOT FOR.
I remember good things about Outlaw Star, though admittedly I haven't seen it since I was a kid.
Another one I liked was Big O, though that show is apparently half-American, so there's that.
Oh yea, and Ghost in the Shell kicks major ass for both seasons and it's movies, so get that.
EDIT: Also forgot to mention Fullmetal Alchemist. While I've only seen the original anime series (it's one of my personal favorites), you may also want to check out FMA: Brotherhood, because from the few episodes I've seen on Adult Swim, it ain't bad either.