What I'm trying to do is watch some of the best Asian films from the past few years, so try to keep your suggestions, at least for now, within the past 5-10 years or so. To give you an idea of what I've already seen in the past few weeks, here you go. Try to come up with something in the same vein.

Hero

Zatoichi

Oldboy

Battlefield Baseball

Attack the Gas Station

Memories of Murder

JSA: Joint Security Area

Sympathy For Mr. Vengeance

You'll get some responses here, but you could also head to the International forum where you will not only find stickies that help answer this question, you will find this question has been answered many times over.

Battle Royale

I know about the stickies, but only one of them would really help me out (the HK thread), and that hasn't been updated in a while now, so for more recent films it isn't very helpful.

:/ Sorry I haven't updated it lately. Bleh.

King of Masks
Shiri
Fulltime Killer
Kikujiro
Failan
My Sassy Girl
Time and Tide
Suicide Club
The Foul King
Versus
Blue Spring
Tell Me Something

Bichunmoo
The God of Cookery
Guns & Talks
Fallen Angels
Koriyoshi Ichi (Ichi the Killer)

Shaolin Soccer
Akira

Since you mentioned "Hero" and "Zatoichi", maybe consider:

Twilight Samurai (Japan)
When the Last Sword is Drawn (Japan)
Warriors of Heaven and Earth (China)
Musa (Korea)
Legend of the Evil Lake (Korea)
The Legend of Suriyothai (Thailand)
Azumi (Japan) - haven't watched it yet myself

I enjoyed "Zatoichi (2003)" but I'm nowhere near as high on it as others seem to be. I liked the first 6 films mentioned above more than "Zatoichi (2003)". Definitely check out the older Zatoichi series if you haven't already.

As for other suggestions:
Save the Green Planet (Korea)

Yi-Yi.

I'm looking for some recommendations of "must see" contemporary Asian films. I've gotten my feet wet, but I'm wanting to expand my horizons a little. What are the essentials? I'm really looking for stuff that hasn't hit US theaters, because otherwise I've probably heard of it or seen it (ie Hero, Crouching Tiger etc).

Two that I have seen recently that I really enjoyed are Oldboy and Save the Green Planet. Anything similar to those two would be what I'm interested in.

I also love Samurai movies. I bought Zatoichi (Takeshi version) and enjoyed it. So anything along those lines would be good too.

And don't worry about Takashi Miike. I've had my fill of that.

And I'll take horror suggestions too, but I have seen a lot of them (Ring, Ju-on, Tale of Two Sisters, etc.) The whole "scary white-faced ghost" theme is beginning to wear thin to me. Are there any good Asian horror movies that don't use this convention?

Oh, and don't feel limited to recommending only ones that are available on Region 1 DVD. I've got a region free player.

Thanks for any recommendations.

Good thread and I was about to start something like this myself.

I have just recently been taken over by Asian cinema and freakin' love it.

Started with Audition, then Dark Water followed by A Tale of Two Sisters and Oldboy.

Caiman - I would reccomend you check out Infection it's a pretty creepy horror flick and I dug it a lot - not the best but pretty damn cool (and no, there are no pale faces covered by dark hair to be found....just a lot of weird shit).

Here are the movies I have lined up that are suppose to be damn good:

Suicide Club/circle
Battle Royale - Link to Arrow's review....spoilers
http://joblo.com/arrow/reviews.php?id=219
Phone
The sisters

I hear Battle Royale is damn good and I don't know much about the others.

Thanks. I've seen (and own) Battle Royale. It is indeed a good movie. Infection seems interesting. I'll add it to my netflix Queue.

Quick note: I just read that Infection is part 1 of a trilogy. That's actually great news because.......well, I'm not going to spoil anything. Let me just say that the first film came off as a perfect fit for a trilogy.

What? The International DVD forum not good enough for you?

That place frightens me.

What's so scary about the "International Forum"?

Does all the talk of being region free make you lose control of your bladder or something?

No, not the bladder. Close though...

Infernal Affairs Trilogy
A Beautiful Boxer
Friend

Cure - Nice murder mystery

Three Extremes 2 - 3 shorts by 3 of Asia's greats (Takashi Miike, Fruit Chan, Park Chan-wook).

Infernal Affairs - Great story about an undercover cop and a mole in the police department.

H - A serial killer is in jail but his murders still continue.

Sharkskin Man & Peach-hip Girl
Fulltime Killer
My Sassy Girl
Blue Spring
The Foul King
Kikujiro
Fallen Angels
Oasis

I'll see your Kikujiro and raise you a Sonatine and Hana -Bi, can't go too far wrong with a bit of Kitano.

If you don't mind non linear narratives along the likes of David Lynch, try most features from his Asian counterpart Shinya Tsukamoto...granted you may not like the likes of Tetsuo but i'd still bracket them under 'must see'

I've been hearing good things about Tetsuo but what's it all about? The write up I read sounded a little wierd (dudes body is meshed with metal and skin or something).

Is it a pure sci-fi flick or are there some horror elements thrown in the mix as well?

Mixture of both really, and not easy to explain (the body meshed with metal and skin sounds right..lol) in essence hence why i likened it to David Lynch's output and non linear narratives.
Shinya Tsuakamoto's films are ones that are going to polarise opinions, you'll either love them or hate them and of the films i've seen of his i'd say Gemini is probably the most accessible although i'm still to see Vital which i'm hearing good things about and is certainly not as weird/outlandish as the likes of Tetsuo.

Hana-Bi is easily one of the best films to ever come from Asia. Sonatine is good, but doesn't rise to the same level. I'd recommend all Beat Takashi's films except for Boiling Point which is interesting but most definitely NOT a good movie.

If you enjoy gross movies from a violence standpoint then I'd highly recommend Riki-Oh: The Story of Riki. It's a great movie to watch at a party because it's totally insane.