I have a Gigabyte OC GTC 680 card paired with a Asus P8Z77-V deluxe board and a 3770K processor. I have the same issue with it not working using PCI gen3 in the first slot. when set to gen2 there are no problems at all.
There have been reports of the same issue with the same solution using IVB i5s and also with Z77 boards from other manufacturers, so I don't think it is an issue with the i7 or with the Asus MBs.
No other card reports this issue, only the Gigabyte OC 680 and 670s (which are basically the same).
I also tested my video card in the second PCI3 slot on my motherboard. That one runs fine at gen3 speeds, but uses 8 lanes instead of 16. My feeling is that the Gigabyte boards would probably work fine in SLI because of that (both slots drop to 8 lanes in that configuration).
I also was using a HD7870 card on my system before I got the 680 (the 7870 was "borrowed" from another computer I have). That card is also PCI3 capabale, and it ran with no issues at all. It did show a similar problem (constantly crashing the driver) when I first got it on my older system (a Dell XPS studio 9100 with a i7 930 based board), but that was fixed with the AMD 12.4 driver release.
So, I think that it is fairly clear that there is something wrong with the Gigabyte OC 680 and 670 cards. Perhaps it is a not a big issue since the cards seem to work fine at gen2, but is still an issue none the less. The cards do not work the way they are supposed to.
Hopefully it can be fixed by a BIOS update (btw, the F2 update does NOT fix the problem).
I also noted that someone here with a Gigabyte Z77 MB doesnt seem to have the problem. That may be the issue, Gigabyte have tested their cards with their own motherboards, and perhaps they work as expected there, but not with other manufacturers motherboards. If Gigabyte are not implementing PCI3 correctly, that would explain it, since the same error would be replicated in both card and motherboard. Interestingly, it might mean that the Gigabyte MBs might not be fully compatible with other PCI3 gear if that is the case.