I can't believe nobody has mentioned Grave of the Fireflies (http://www.imdb.com/title/tt0095327/) yet. Granted, it's anime, but it still has as much emotional depth and drama as any of the better asian films; probably one of the best.
My favorite is To Live (http://www.imdb.com/title/tt0110081/), which I also can't believe anyone hasn't mentioned yet. It's about a man with a gambling problem who reforms himself, and after losing everything due to his gambling, his family must now also persevere through the Communist revolution in China. It sounds like a played out storyline, but if it is, it's the one with the best execution. Everything about this movie is the awesome. If you're really looking for an asian film that definitively embodies the scope of asian cinema, check out this classic. You'll cry, you'll laugh, and you'll feel heartwarmed. If anybody else has seen this movie, vouch for it so that I don't look like a homo.