Title pretty much says it all. I love the charm of Japanese games but I'm no fan of turn-based combat. The Kingdom Hearts series is one good example that I have played and loved. I'd also be willing to give non-RPGs a chance.

Dark Cloud 2

Star Ocean series

Okami

Legend of Zelda series

Demon's Souls and Dark Souls.

Catherine (not a JRPG, by any stretch of the imagination; it's a cutscene-heavy puzzle game with a plot)

EDIT--

And, if you're able to find them...

Megaman Legends series

Steambot Chroncles

If you want another Square-Enix game that is somewhat similiar to Kingdom Hearts as far as gameplay goes, I highly recommend http://en.wikipedia.org/wiki/Musashi:_Samurai_Legend. I actually enjoyed it more than Kingdom Hearts. http://en.wikipedia.org/wiki/Threads_of_Fate is very good too (and easily available on US PSN).

If you want a more recent game, then I highly recommend http://en.wikipedia.org/wiki/Nier_(video_game) which was released for both PS3 and 360. Brilliant game, possibly my favorite of this console generation so far.

Ah, yes, excellent game. Shame the sequel for the PS3 been cancelled

What about the first game?

It's alright, but the sequel is an improvement in pretty much every way.

If you've got a DS, I can give http://en.wikipedia.org/wiki/The_World_Ends_with_You a very strong recommendation. It was co-developed by the Kingdom Hearts team, and the combat's a lot of fun, especially once you've found a pin combo that works well for you.