With a lot of JRPGs and Japanese SRPGs, even when they have great reviews, what I hear about the stories is not very impressive. Intriguing and fun mechanics are great, but it would be nice to have a story that further motivates you to play .
I think I could name a few JRPGs with stories that I enjoyed. I know there are those who hate Kingdom Hearts with an extreme passion, but the story in the first game wasn't bad...really. Radiant Historia is another one that's memorable. I would argue that the story in Valkyria Chronicles was at least decent, but I know a lot of you did not feel the same way . The music accompanying the gorgeous cutscenes definitely helped in that one (spoilers).
And of course there's P3 and P4.