I guess I'd be in the minority in that Chrono Trigger and Xenogears didn't really do much for me. I've certainly played worse games, but they just didn't come anywhere near to living up to all the hype people throw at them IMO.
That said, the SNES-PS2 era was kind of the golden period for JRPGs in a lot of ways. Final Fantasy 7 really kind of ushered in a new era of a more cinematic experience, and IMO, FF8 largely improved on the few areas lacking in 7. It stumbled in its own right in a few places, but overall I considered 8 to be the better game. FF9 was probably worth at least one playthrough, but not sure about anything more than that.
The first Grandia game for the PSX (well, technically Dreamcast) was pretty good, and had kind of an interesting combat system where if you timed your attacks properly, you could delay the attacks of enemies. The second Grandia for the PS2 was serviceable, and the third has the feel of the game being maybe 75% done and then the developers had to try and stitch it together as best they could to shove it out the door. The mechanics are all fine, it's the plot that has rather large gaps in it.
Tales of Symphonia was a great gem for the GameCube. There was a PS2 port, but I don't think it ever made it out of Japan. Maybe not quite a JRPG, more of a Resident Evil kind of game, but Eternal Darkness was another little gem for the GameCube.
I know there will be plenty who disagree with this one, but I think Final Fantasy X is kind of the pinnacle of modern JRPGs. While it is certainly the case that you're on a very linear track, the story is what is driving the game, and the story is kind of the whole point of a JRPG. However the combat in FFX is great as well. Turn based, none of that stupid active time battle crap, and you can swap characters in and out on a whim depending on the situation. And then the story in FFX has a little bit of everything. Love, loss, religious intolerance, political intrigue, a bad-@ss monster, and you're even drip fed about 1,000 years worth of history for the world. I could have done without Blitzball, but you really only have to play one game, and it's not required to win.
Xenosaga was also a pretty good trilogy for the PS2. IIRC, it's intended to be kind of a prequel to Xenogears, and while it sadly got cut short from 6-8 games to 3, it still packs in an ungodly amount of content.
Tales of the Abyss was also an excellent PS2 RPG. Tales of Legendia was serviceable if a little uninspired.
You can pick up used GameCube and PS2 units on the cheap these days, so I'd suggest getting at least a PS2 so you can then also play a lot of older PSX games that may not have been released on PSN for the PS3.
On the 360 and PS3 front there's Tales of Vesparia which was a fairly solid, if somewhat unremarkable, title. Tales of Graces f is another solid, if kind of ho-hum kind of game (IMO). Final Fantasy XIII was a great game IMO... Also very linear like X, but again the story is driving things. XIII-2 tried to be less linear and seemed to lose the cohesion of the story you had with XIII. You could be wandering around for hours before hitting on the next segment of the story arc, whereas XIII had a pretty steady dolling out of story the entire game.
Eternal Sonata was probably a solid B level game. If you're a Chopin fan, you'll probably love all the little references in Eternal Sonata. Star Ocean 4 was serviceable if rather forgettable.