The guy who directed Perfect Blue did a show called Paranoia Agent, which is a little depressing but has some great single episodes, and a movie called Paprika, which is weird but amazing.
The guy who did the Girl Who Leapt Through Time just released the movie Summer Wars in the states. Excellent film with much the same tone. He also did the sixth One Piece movie, the Baron of... something or other, which is... different.
I haven't really seen anything that compares to Death Note, but a friend recommended a similiar epic crime drama called Monster. I didn't love it; although it had some good moments throughout, the ending was disappointing.
The Akira director, Katsuhiro Otomo, did a steampunk film called Steamboy, and a collection of short films published as "Memories".
Lets see, what else.
Eden of the East is slightly surrealistic story about a contest between a dozen people to "save the nation of Japan." Great mix of traditional animation and cell shaded CGI. It ran for one season, then wrapped up with two movies.
Moribito: the Guardian of the Spirit ran on Adult Swim a few years ago. Excellent fantasy series about a gifted warrior woman hired to protect the son of the local emperor after his father orders his execution.
Mushishi is a stunning fantasy/horror series that I would call a quiet, naturist version of the X-Files set in remote areas of early 1900s Japan.
Katanagatari is about to be released in the states. Beautifully stylized, it follows a master strategist and an unarmed fighting expert on their journey to secure twelve evil magic swords. Excellent from start to finish.
Night Raid 1931 (originally Senkou no Night Raid) is an interesting "secret history". Set in Shanghai, it follows a Japanese army spy team with super powers as they uncover a plot to change the course of World War 2. I think this is also soon to be released.
And not yet released over here, but available as fansubs:
Occult Academy
Real Drive
Stein's Gate (still running in Japan)