Ok so here's a little review tour of the most prominent examples of Asian cinema:
Oldboy - Chan-wook Park - 2004 (S. Korea)
Pretty much the ultimate revenge film, the second part of the Vengeance trilogy (the the first part is Sympathy for Mr. Vengeance, the last part Sympathy for Lady Vengeance), it's a story of a man that gets locked up in a room for 15 years for apparently no reason. He suddenly gets released and all sorts of diabolical strangeness ensues...
3-iron - Kim Ki-Duk - 2004 (S. Korea)
This is Ki-Duk's most famous and best film, it revolves around a young "burglar" who breaks into people's houses but steals nothing, and a battered housewife who he encounters while breaking in to her house after which they begin a romantic relationship. The film is almost entirely without dialog, the main actors never speak. Despite this fact, this is one of the most romantic and magical movie experiences I can remember.
Musa The warrior - Sung-su Kim - 2001 (S. Korea)
A semi-historic Korean epic, one of the biggest and most expensive films in Korean history, it's a story of a Korean delegation to China in the 14th century that tries to get back to their homeland after they have been mistaken for spies and chased by the Mongolians. There's not wuxia style fighting in it, but it is brimming with great action (although more rooted in reality).
Spring, Summer, Autumn, Winter... and Spring - Kim Ki-Duk - 2003 (S. Korea)
Second best Kim Ki-Duk film, it's set in a Korean Buddhist monastery, located on a beautiful mountain lake, and revolves around a monk and his disciple, and the various life lessons they are taught while the seasons change. An immensely beautiful and meditative piece of film.