Frankly,
current-gen games are generally very light in the
story-driven genre.
If I had to suggest a few...
Assassin's Creed series, though I'd just skip the first one as it is nowhere near as ambitious/epic as the rest of the games in the series. Play it on the PC, unless you are wed to console controls (though you can use a gamepad on the PC as well).
Witcher/Witcher 2. Gameplay can be a barrier of entry for the first game, but these are also solid (and 'mature'), story-driven RPGs. The first game isn't really current-gen, but the second most definitely is. The games essentially continue Geralt's story from the series of novels, and it helps your appreciation a lot if you read them (or as many of them as you can find) first. Play them on the PC.
Tales of Vesperia is a solid JRPG on the Xbox 360. It's not amazing, but it's solid. If you've followed the lackluster JRPGs available for that system, then 'solid' should be more than enough.
Lost Odyssey is an old-school JRPG in every sense of the word, with current-gen graphics. Don't play this if you're looking for something revolutionary, but if you love the heck out of old-school, story-driven JRPGs of the SNES/PS1 era, then you should find plenty to enjoy out of Lost Odyssey. Plus, the memories/dreams you can unlock are some of the finest short stories in all of gaming, with the only downside being none of them are voice acted.
Dragon Age Origins and DA 2. They're fairly derivative of pretty much every Bioware game ever made, but there is a lot of lore behind the rather staid story of the first game, and if you have a dark sense of humor, the second game's over-the-top plot can be quite amusing, as well as whether or not you're able to realize the real story going on behind the one that seems to be going on (if that makes any sense). DA 2 is probably the worst offender I have ever seen in recycled areas, but if you can somehow ignore that, it's actually a fairly solid game; though the lost potential as you realize what they were forced to cut in order to finish it in 15 months can make you howl in frustration (as opposed to the 4 year development time of the first game).
Red Dead Redemption. Xbox 360. If you enjoy sandboxes with solid stories (ie: most Rockstar games), then unless you just have an aversion to cowboys and westerns, you should enjoy the heck out of this game. Most sandbox games don't have stories worth a damn (ie: Skyrim), but Rockstar almost always makes sure there's a story worthy of a story-driven game in their sandboxes, and they continue to deliver with Red Dead Redemption. For the record, I don't enjoy sandboxes, as I'm primarily a story-driven gamer.
And, that's about all I can think of when it comes to story-driven games of the current gen of PCs and the Xbox 360.
For prior gen games:
Xenosaga series. On the PS2. It's pretty much the most epic space opera in video game form ever made, though it was cut short due to poor sales. It is also almost the very definition of story-driven, as gameplay is just tacked on to drive the story. But, oh what a story.
Metal Gear series. The action/stealth series also has one of the best storylines in video games. They're relatively short, as action games tend to be, but they're also one of the very few action game franchies out there with a real/deep story (as opposed to something just tacked on).
Zone of the Enders games. Sci-fi/mecha with a fantastic story. Another rare story-filled action series. No surprise it's done by the guy who gave us Metal Gear.
The Last Journey/Dreamfall. TLJ is the best story-driven adventure game ever made (though I feel the recently released The Secret World-despite being proclaimed an MMO-is a worthy spiritual successor to TLJ). The graphics are definitely dated, as this game came out over ten years ago, but the story is still as great as it ever was, and April Ryan puts pretty much every other female video game protagonist to shame. Dreamfall was the rather awkward sequel, and is worth playing just to continue the over-arching story, but was disappointing in that it concentrated on a new protagonist, and didn't have any real conclusion as it was originally envisioned that the game would have chapters to tell its whole story, but poor sales put a kaboosh on that.
Xenogears. Spiritual predecessor to the Xenosaga series. Probably the best story-driven game on the PS1. Lots of mecha, religious/philosophical musings as only the Japanese can do, etc. Disk 2 is probably the worst offense ever in gaming history, but you'll spend so much time on Disk 1 that you'll force yourself through Disk 2 so as to complete your journey through this epic classic.
Shadow Hearts and Shadow Hearts: Covenant. On the PS2. These games have a prequel of sorts, called Koudelka, on the PS1, but you may find the gameplay frustrating. Story-wise, it's pretty great though, and adds some background to the Shadow Hearts games. They also have a third game in the series, but it is barely related to the first two Shadow Hearts games, and its story isn't particularly enticing (though gameplay is probably the best of the series). The Final Fantasy games made all the splash when it comes to JRPGs for the PS2, but these two games are the best story-driven JRPGs (mixing demons, religion, and fictional history of the World Wars) for the console (behind Xenosaga), in my eyes. The first game is rough behind the ears, but Covenant polishes it and pumps out a 40+ hour epic. Unlike most story-driven games, you do need some gaming skills (ie: reflexes), to some extent, as combat is timing based, and I can't help but feel that's the reason these games never met mainstream JRPG love.
I'm going to stop here, as the list of old-gen story-driven games could go on for pages, as those games couldn't rely on eye-candy to drive sales. Hopefully you're able to find something enticing in the above suggestions.