Although it's not an RPG, Hakuoki: Demon of the Fleeting Blossom is a very good PSP visual novel. Lovely anime art, cool music (love the few vocal themes), branching pathways based on character interactions, but nothing in terms of dungeon crawling or world exploration.
Another non-RPG you might like is Kenka Bancho: Badass Rumble. There's lots of exploring, NPC interaction (whether to talk or pick fights), cool art, music that may suit you, branching pathways... Yeah, I definitely think you should check out Kenka Bancho: Badass Rumble.
I definitely +1 the recommendation for Trails in the Sky. Excellent RPG.
I would vehemently recommend AGAINST Mimana Iyar Chronicle. That game was an absolute chore to play. I didn't like it, but there are a few who did. And what do I know? I also disliked Fate/Extra. (My reviews of both are in the site's reviews section.)
And are you looking only for PSP-only games, or are older RPGs on PSN fair game? I happen to think Legend of Dragoon is a solid JRPG (especially the music), but I'm in the minority there.