I'm a big fan of Japanese movies, from anime to Kurosawa. But I recently went looking for a change of diet and discovered some great Korean movies:
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3 Iron, about a transient who lives in the houses of people on vacation.
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Oldboy, a fantastic, kinetic movie about a man who is kidnapped and imprisoned for 15 years and then inexplicably released.
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Spring, Summer, Fall, Winter... and Spring, a beautiful, stately (read: slow-moving) Buddhist parable.
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Tae Guk Gi, The Brotherhood of War, a (somewhat melo)dramatic Korean War epic in which brothers fight on opposites sides.
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The Host, a funny, offbeat modern monster movie with an environmental, antiestablishment bent.
Of the bunch, I'd say "The Host" is the most accessible, but "Oldboy" was my favorite.