@Zloth BeardyHat is exactly right. JRPG stories that I've been inflicted with have all been dreadfully painful and very hard to endure. That being said, I have mostly avoided them due to that reason and so I don't have as much experience as you do with them. If you have suggestions for good ones, I'd be more than happy to give them a try.
I would have previously described myself as a JRPG hater, but after reflecting back over the last couple of years, I've realized that I've actually played
and enjoyed quite a few of them, though rarely completed them. I've come to the conclusion that I just hate Final Fantasy as a group of games, though I do generally regard JRPG combat as pretty boring, but I can sit through a lot of it, apparently. So here's a short list of the ones I've played and enjoyed, in no particular order
Felvidek - Just played this this year and it's not really a traditional JRPG in that it's made by Eastern Europeans and set in 15th century Slovakia. It also doesn't have random battles, something I generally dislike in the genre, but the story is decent and the characters are full of depth and interest, plus it's generally just hilarious. It's only about 6-hours, so I
highly recommend it. It was one of my favorite games this year.
Dragon Quest - I've played quite a bit of Dragon Quest, including 50-hours of 8, completed 4, started and played about 10-hours of 5 and am currently working through 3 on Gameboy Color. What I generally like about Dragon Quest is that it foregoes the melodrama of Final Fantasy and tries to tell a fairly basic story and isn't constantly hitting you over the head with dialogue and cutscenes. In fact, in the earlier games, cutscenes are pretty few and generally quite short. 8 has more of them and has more characters in it, versus something like 3 where you are the Hero and then create a party around the hero of pretty much blank faced members. That said, the characters in 8 are fun and characterful, maybe a little over the top. I also very much appreciate Dragon Quests' music, sense of humor and art style. People do generally regard the stories (especially 1-3) as having quite a lot of depth and interest, but frankly, it's not something I really explore.
Persona 3 Portable - I'd like to play the other Persona games, but I feel like I need to finish this one. That said, after something like 70-80-hours, I am dead bored of the combat, which there tends to be a lot of. The best part of the game is the social stuff; you're a teenage high school student and you have a limited amount of time in a day, so you need to choose what to do with it. You're generally obligated to attend class, but you then have to choose who you're going to hang out with and where you're going to go in the intervening hours. I find the story nothing to really write home about, but the characters are pretty fun to get to know and fun to hang out with and build bonds with, which help you out in combat situations. There's definitely some generic "Anime-Stuff" early in the game, but I feel that it mellows out and goes in different directions after a dozen hours or so. This is very much a different style of JRPG if you're mostly familiar with Final Fantasy or Dragon Quest.
Yakuza - Like a Dragon - This has to be one of my favorite games in recent years. The story is not about saving the world or felling the ultimate bad guy, but instead follows (pretty much) a team of losers as they fight to investigate Yakuza and other gang dealings within their own little world. Ichiban, the protagonist, has to be one of the best characters to grace the video game world in decades; he's genuine, kind and absolutely hilarious. He's obsessed with Dragon Quest as a game series and views almost everything in his world through that lens; his bat that he beats people with? That's the heros weapon with which he uses to fell evil. The game has an absolutely dark sense of humor; I've never played a game that is just so absurdly funny to me. For example, almost every RPG has mimics, right? Well, Like a Dragon has those as well, but instead of Mimics, they're "Degenerates" who hide in safes and closets, jumping out to attack you. You beat-up "Perverts" and get attacked by the "Hungry-Hungry Homeless." Yeah, it's not exactly...sensitive, but it's extremely funny and surprises me constantly.
There's just a few suggestions. Generally I think I favor character over story, though the Dragon Quest series is more lacking in that regard, but I tend to at least enjoy the gameplay and the world building there; plus I just find the rest of the game so damn charming, it's hard to not love.