** this list does not include Anglo-countries.
1. Hero (2002): A love story pretending to be a martial arts film, this Jet Li film boasts amazing scenery and imagery, strong characters and a strong story. Watch this on a big screen.
2. Diva (1981): This is a fascinating film about an obsessed French mailman who steals a dress from a washed up Opera singer and then returns it to her while he finds himself being chased by the mob and corrupt cops. Of all the films on the list, this one feels the most foreign and it's kind of the most interesting... in an odd sort of way.
3. Tampopo (1985): Done in the style of a spaghetti western, this film involves a truck driver who comes to town and teaches a woman to make the best noodle restaurant in the area. This is another one of those films where you get real cultural differences and a surprisingly funny and heartwarming film. By and large, this film is very accessible, though there is a narrator (a hedonistic gangster and his girlfriend) who feels very out of place.
4. Ikiru (1952): Akira Kurosawa is perhaps the greatest director of all time and this is his most touching film. This is the story of a lowly bureaucrat who learns he is dying. He decides to do what he can to get a playground created against the wishes of "the system." His funeral is infuriating. The film quality is really poor because this was filmed in low quality black and white in post-war Japan, but this one is absolutely worth seeing.
5. Das Boot (1981): Starting life as a miniseries, this was recut into an unmatched film about the realities of submarine war. Directed by Wolfgang Petersen and starring Jurgen Prochnow, you can't help but feel the pride, joy and ultimately horror these men experience as they struggle to survive impossible odds. If you ever want to imagine how it would feel to die at the bottom of the ocean, this is the film for you.
6. High and Low (1963): I could fill this list with Kurosawa films, but I'm refraining. Apart from Ikiru, this is my other personal favorite. Toshiro Mifune, one of the best actors ever, plays a shoe company CEO who learns his son has been kidnapped. Things change for the worse when he learns that it was actually the son of his employee who was kidnapped and everyone still wants him to pay a ransom that will bankrupt his company.
7. Lola rennt (Run Lola Run) (1998): Starting Franka Potente as Lola, this film involves Lola receiving a call from her boyfriend, who is desperate for cash and is about to rob a grocery store to get it. She runs to stop him. She fails. So the film resets and she tries again. With each pass, we see alternate histories for the characters presented in a series of snapshots. This is a fascinating and bizarre film. Unfortunately, there are some serious mistranslations in the film that might confuse you (it's best to speak German), but it's still well worth seeing.
8. Jean de Florette (1986): Gérard Depardieu plays a likable man who moves his family to the country only to have a rotten neighbor sabotage his efforts at farming. Yeah, that doesn't sound like much, but the film really works.
9. My Sassy Girl (2001): You know how in every rom-com the heroine will be a bitch (in a very safe and cliche manner) and drive away the male love interest, only to have him return when he realizes that he does love her and that he accepts her the way she is? Aw. Oh course, we also learn that the heroine is not really a bitch because no American starlet would accept such a part as they all want to be "America's sweetheart," so her bitchiness is written off as a mistake or misunderstanding. Well, not here. What I love about My Sassy Girl is that for once, the heroine really is being a bitch and she's trying hard to drive away this guy. He just won't leave and they fall for each other. The ending is heartbreaking too.
10. La Femme Nikita (1990): It's hard to say if this should be included because, while it is French, director Luc Besson would soon make it big in America and this film feels very Hollywood. This idea was also imported here for a film and a television series. Still, the original is quite good.
Honorable mention to Black and White in Color, in which a bunch of bumbling colonialists decide they need to go to war since Europe has gone to war, to all the Japanese horror films that got copied and became The Grudge, The Ring, and Dark Water, and the Korean The Wishing Stairs, which is a horror story in a girls school but strangely ends up as a solid lesbian romance movie. Yeah, weird. You can't go wrong with Godzilla either.
I do wish I liked Lost in Translation better.