Depends on what you mean by artsy? If you mean avant garde or unusual or like what we would think of as indie films in the US, I can't think of many dramas like that, but I would recommend for movies:
Castaway on the Moon (one of my favorite all time movies)
I'm a Cyborg, but It's Okay (probably one of the 5 single weirdest movies I've ever seen and I like weird movies, plus Rain, so win!)
Late Autumn
Oishii Man (I don't know if this is actually any good, even after seeing it, but my bias for Lee Min Ki is so great, I love anything where I get to see him)
The Railroad (won the FIPRESCI award and Kim Kang-woo won best actor at the 25th Torino Film Festival, BUT it's really hard to find with subtitles in English. I had to download it from Asia Torrents, but it was totally worth it.)
I guess the most avant garde looking drama I've seen is White Christmas. It's pretty eerie and beautiful and weird (at least so far, I haven't finished yet).
Painter of the Wind is beautiful in terms of being artistic, about art, and aesthetically pleasing, but not sure if that is what you mean.