It turns out the answer to this is quite simple: upgrade the BIOS. The motherboard has a built in graphics adapter and VGA output, so if you need to you can use that to boot your machine up, but you'll first have to remove anything in the PCI-E slot.

Then visit Gigabyte's website and download and run @BIOS, choose to "update BIOS from gigabyte server" and it should find the most recent stable version. For me that's version F22.

After completing that it should boot correctly with the new card.

Note that:
1. Updating your BIOS is slightly risky in that if it goes wrong your motherboard won't might not work anymore, so you should only do it if your system really needs it to work correctly.
2. After updating it will probably have reset all your BIOS settings, so if you've chan[...]]]> After upgrading my graphics card to an NVIDIA GTX 750 Ti(not really intentionally, the previous one died), I found that my GA-Z77-D3H motherboard (with BIOS version F13) froze on the boot screen.

It turns out the answer to this is quite simple: upgrade the BIOS. The motherboard has a built in graphics adapter and VGA output, so if you need to you can use that to boot your machine up, but you'll first have to remove anything in the PCI-E slot.

Then visit Gigabyte's website and download and run @BIOS, choose to "update BIOS from gigabyte server" and it should find the most recent stable version. For me that's version F22.

After completing that it should boot correctly with the new card.

Note that:
1. Updating your BIOS is slightly risky in that if it goes wrong your motherboard won't might not work anymore, so you should only do it if your system really needs it to work correctly.
2. After updating it will probably have reset all your BIOS settings, so if you've changed your disk boot order don't be surprised if it boots the wrong one.]]>