I humbly consider myself an expert on the topic.
So here is my "starters kit":
Korea
Watch everything by Kim Ki-Duk (*www.imdb.com/name/nm1104118/), if you haven’t already. It’s a bit "arthouse" but still worth a try.
Also, check out Park Chan-wook (*www.imdb.com/name/nm0661791/) his "Sympathy for Lady Vengeance" and "Sympathy for Mr Vengeance" form a trilogy with "Oldboy".
Thailand
Definitely check out Pen-Ek Ratanaruang (*www.imdb.com/name/nm0711579/), "Last life in Universe", "Ruang Talok 69", "Invisible Wave" are great pieces of work.
China
Wong Kar Wai (*www.imdb.com/name/nm0939182/) is very heavy to get into at first but it’s wonderful to watch especially his early low budget stuff.
Zhang Yimou (*www.imdb.com/name/nm0955443/) now known for "Hero", "House of Flying Daggers",.. he started of in something completely different. "Raise the Red Lantern" is a fantastic classic.
During the late 90s a lot of underground films were made under poor conditions always running from the authorities showing pictures nobody should see, the films were smuggled out of china and developed by European TV channels. That’s something every body should look into. a few titles: "Beijing Bicycle" (*www.imdb.com/title/tt0276501/) is probably the most known. "Suzhou River" (*www.imdb.com/title/tt0234837/) and Ke Jia Zhang’s movies (*www.imdb.com/name/nm0422605/)
Hongkong
Well Donnie Yen (*www.imdb.com/name/nm0947447/), Johnny To (*www.imdb.com/name/nm0864775/) if you’re down for intelligent action films. "Beyond Our Ken" (*www.imdb.com/title/tt0440445/) is a movie you can easily watch with someone who isn’t into asian movies AT ALL.
Taiwan
Check out early Ang Lee it’s wonderful. Hou Hsiao-hsien (*www.imdb.com/name/nm0396284/) is a great story teller ("Millennium Mambo" or "Café Lumière" )
Japan
Takeshi Kitano should be in everyone’s collection. Toshiaki Toyoda (*www.imdb.com/name/nm0870313/) did great movies until he lost his career due to 5gr of Marijuana.
If you’re really into very slow moving stuff, into real people in real situations then here’s a little tip: Naomi Kawase (*www.imdb.com/name/nm0442905/),Hirokazu Koreeda (*www.imdb.com/name/nm0466153/) and Shinji Aoyama (*www.imdb.com/name/nm0031888/) if you’re into this you will be amazed.
Late Shohei Imamura (*www.imdb.com/name/nm0408076/) basically did the Japanese Nouvelle Vague.
If you’re into more extreme stuff, you already know Takeshi Miike (*www.imdb.com/name/nm0586281/) and Sogo Ishii (*www.imdb.com/name/nm0411025/), if you want to take it further watch Shozin Fukui (*www.imdb.com/name/nm0298006/)2 movies.
A movie most positively every one with a functioning mind will appreciate would be "The Taste of Tea" (*www.imdb.com/title/tt0413893/).
Honorable mention in the wth-category goes to Sion Sono’s "Strange Circus" (*www.imdb.com/title/tt0492784/)
So yeah, this would be a light scratch on the surface of Asian Cinema. It has everything and it’s my firm believe that is greatly superior to Hollywood and most of European Cinema.