I'm surprised some of these weren't mentioned.
The Seven Samurai - A classic tale about a village being ravaged by bandits. The peasants, who hate the samurai, set out to hire samurai willing to fight for them with food and boarding as their only compensation. An amazing film that inspired films such as The Magnificent Seven and Seven Swords as well as the anime Samurai Seven.
The Musashi Trilogy - The story of the legendary Miyamoto Musashi ranging from his early life to his duel with Kojiro Sasaki.
Fighter in the Wind - I'm pretty sure this movie was made in Korea. The story centers around Kyokushin founder, and former Toughest Man Alive, Mas Oyama's life as a Korean stow away looking to become a fighter pilot in Japan. Instead he revolutionizes the martial arts world defeating some of the toughest Martial Artists in Japan.
Once Upon a Time in High School: The Spirit of Jeet Kune Do - The story about a high school student in I think the mid-late 70's. He attends school in a totalitarian-esque high school and befriends another guy known to be an excellent fighter. When stuff goes down, the main character will decide enough is enough and begins training in Jeet Kune Do, to finally stand up for himself.
Oldboy - Not exactly a martial arts film, but it has one of the best fight scenes ever.<- Period.
Now one thing that I have to add is that I'm a huge fan of the Korean fight choreography. The fights are the most realistic I've seen in a long time. There aren't too many of the old Hong Kong fight formula which is: Main character beats up all the normal guys no problem. Fights the tough guy and gets beaten up a bit, but eventually wins. Then fights the main villain, gets beaten up a lot. More dialogue, and then finally the main character defeats the main bad guy. Seriously, just watch the flicks. You'll like them.