PCIe Generation 3.0 graphic board running on old mainboard Generation 2.0 or earlier?
Solution 1:
They're usually fully backwards compatible, given your PSU is powerful enough. Just yesterday I installed a new PCI-E 3.0 card on my old PCI-E 2.0 board and had absolutely no issues. Sure, it won't run at it's maximum performance, but it works till I'll replace the mainboard.
However, some of the earlier specs (I think 1.0 and/or 1.1) might have issues running a PCI-E 3.0 so card. I might be wrong here, but I wouldn't recommend doing that. But using the card on a 2.0 board is perfectly fine.
Solution 2:
It works just fine, i have a ATI 7970 on a X58 chipset, and it works fine at PCI-e 2.0. One note tho: sometimes the power management will put the PCI-e bus into 1.1 mode for energy saving, so don't let it freak you out when you see your new and shiny 3.0 at 1.1 ;)
Solution 3:
A PCI Express 2.0 motherboard will support a PCI Express 3.0 graphical card, but it will work on the 2.0 bandwidth.
A PCI Express 3.0 motherboard will support a PCI Express 2.0 graphical card, but it will also work on the 2.0 bandwidth.