Some more recommendations:
Afghanistan
Osama (2003) - Siddiq Barmak
Earth & Ashes (2004) - Atiq Rahimi
Both films are very much done in the Iranian style.
China
Anything by Zhang Yimou is good and his movies are quite "mainstream" I guess. Apart from his more famous ones (like Hero), I enjoyed:
The Road Home (1999) with Zhang Ziyi
The Story of Qiu Ju (1992) with Gong Li
Happy Times (2000)
Riding Alone for Thousands of Miles (2005)
I haven't seen much by some of the lesser known Chinese directors but liked what I saw from Tian Zhuang Zhuang (The Horse Thief) and Jia Zhang Ke (Platform).
Kekexili: Mountain Patrol (2004) - Lu Chuan is also a good movie about Tibetan antelope poaching.
Hong Kong
Watch any Wong Kar Wai film (except perhaps My Blueberry Nights). Seriously.
India
I've really only seen Satyajit Ray films aside from the more popular Bollywood movies. If anyone is unsure about watching his movies, check out Sadgati (1981). Shows his style pretty well and is nice and short - only 45 minutes.
Iran
A Separation (2011) - Asghar Farhadi is one of the best films I've ever seen.
Kandahar (2001) - Mohsen Makhmalbaf. I agree with Bluedeed that Makhmalbaf is weaker than some other Iranian directors but this is worth seeing.
Blackboards (2000) - Samira Makhmalbaf (Mohsen's daughter)
Anything by Kiarostami or Majid Majidi is worth watching.
Japan
So many to choose from but apart from the many famous directors, I'd recommend anything by Tetsuya Nakashima (particularly Confessions and Memories of Matsuko), Takeshi Kitano (Dolls & Fireworks), Hirokazu Koreeda (Still Walking, Maborosi & I Wish) and Takashi Miike (for something a bit more violent
)
Philippines
Yam Laranas has made three horror movies worth seeing (although two are the same movie):
The Road (2012)
Sigaw (2004) Philippines Version
The Echo (2008) US Version
South Korea
Kim Ki Duk - 3-Iron and Spring, Summer ... are masterpieces
Park Chan Wook - Oldboy, Lady Vengeance & Thirst as well as Stoker if it counts
Lee Chang Dong - Oasis, Poetry & Peppermint Candy
Bong Joon Ho - Memories of Murder and Mother
Kim Ji Woon - I Saw the Devil, A Tale of Two Sisters & The Good, The Bad & The Weird
So many great movies out of Korea. Some others:
Breathless (2009) - Yang Ik Joon
Castaway on the Moon (2009) - Lee Hae Jun
Bedevilled (2010) - Jang Cheol So
Taiwan
Watch every movie Hou, Yang & Tsai have made
Thailand
I've liked everything I've seen by Ratanaruang and Weerasethakul. I'd also recommend this - one of my favourite horror movies:
Shutter (2004) - Banjong Pisanthanakun/Parkpoom Wongpoom