1. The Happiness of the Katakuris
This is a murder musical with the odd bit of claymation. A family own an inn. Bad things happen. They sing about it. It's great, The bits with Richard Sagawa (pictured above) are my favourites.
2. April Bride
This film is about a blossoming relationship and breast cancer. It was beautiful. I cried my butt off. Watch with caution.
3. GeGeGe no Kitarō
A 2007 live action film based on the popular 1960s manga of the same name, Kitaro is a goofy story about various supernatural characters. There's a 2008 sequel which is fun too. Eiji Wentz's wig in either film is not good.
4. L: Change The World
I thought this L-based spin off was much better than the Death Note film, but then again I just love L, so you know. He's looking after kids and eating chocolate. I've forgotten the rest of the plot, but that's all you need to know.
5. The Grudge
Sorry to be super obvious but I have watched this film so many times and every time I love it, so I had to include it even though it's got to be one of the most well known Japanese films in existence. I have to say I am a big fan of this style of horror movie. I like the way the ghosts' stories unfold in this as well as the Ring series, Dark Water, and lots of others, but The Grudge is a particularly well contained story. I kinda wanna watch it now, actually.
Other interesting Japanese films include Uzumaki (I wrote a review on it here - the film is worth a watch, but the manga is much better), Hausu, and The Taste of Tea (I haven't finished watching this one, but it's really cool and involves an apparition of a little girl's giant head).