The term" Asian Cinema" covers a range of offerings, from the symbolic and lyrical (Kurosawa) to the downright depraved ("Guinea Pig "series). Ran is excellent, if you're a fan of that style and it is also true to the spirit of the source material (King Lear and Macbeth). The Lady Macbeth character is brilliantly written and acted; pure evil.
I would say that my favourite film by that director (Akira Kurosawa)is [Kagemusha (The Shadow Warrior) . It is a great character study, and the way in which the final battle scene is filmed gives it a profundity and sense of tragedy i've never seen equalled. Speilberg appears as a clumsy amateur in comparison, IMO.
If you have a strong stomach and a particularly dark sense of humour, I would check out films by Takashi Miike, particulalry Ichi the Killer and Audition, but be warned: he makes Quentin Tarantino look like Walt Disney.
Two other Far Eastern films that relly impressed me were Spirited Away, for anime fans and House of Flying Daggers, which is visually stunning. I'd never really thought about production design or art direction before (I don't know taupe from turnips), but HoFD is breathtaking, plus is has all the stylized martial arts and mythic scope a fan of Crouching Tiger, Hidden Dragon could ask for.