Starting with Japan...
Newbies should certainly check out films by Kurosawa.
Seven Samurai and Yojimbo are essential viewing and were remade in Hollywood as Magnificent Seven and Last Man Standing (and in Italy as Fist Full of Dollars). Hidden Fortress inspired Star Wars but is vastly different.
Ran is an epic masterpiece.
Google these amazing directors for more titles
Shohei Imamura (Black Rain-beautifaul yet devastaing look at post-bomb Japan, Ballad of Narayama- kind of like a very rural/ancient Logan's Run
. There is another Ballad of Narayama which is also excellent but I don't rememeber the director)
Nagisa Oshima (his early films are hard to find but are among my favorites. You could probably find In the Realm of the Senses and Merry Christmas Mr. Lawrence (in english and with David Bowie) at most good video stores.
Hiroshi Inagaki (a lot of people rave over the Samurai trilogy starring Toshiro Mifune, to me there too slow, I prefer Chushingura which ain't terribly fast but feels more epic)
Folks who like horror and fck'd-up cinema in general should check out the prolific Takashi Miike (Audition-last 10 minutes will make 97% of people leave the room, Ichi The Killer, Visitor Q, Dead or Alive, and Happiness of the Katakuris-a totally insane musical!)
I highly recommend Onibaba, a black&white ghost story with little or no dialogue, I don't quite recall.
Another good ghost story is Kaidan (or Kwaidan) directed by Kobayashi.
Actor/Writer/Director/Comedian(yes comedian) 'Beat' Takeshi Kitano is an unstoppable force in Japan. His movies are usually quite violent. A few rec's- Violent Cop, Battle Royale, Zatoichi (the orginal Zatoichi films from the 60's/70's are pretty cool too as is the Lone Wolf and Cub series), Fireworks, Sonatine
Greatest anime director ever- Hayao Miyazaki (Spirited Away, Princess Mononoke, My Neighbor Totoro, Nausicaa)
Other essential anime- Akira, Ghost in the Shell, Ninja Scroll, Grave of the Fireflies
Finally, the original Godzilla (the japanese version without Raymond Burr) is one of my all-time favorites.
OK, I was going to list some Chinese films but this will have to do for now, and sorry for all the parenthesis.