I have the exact same motherboard and was going to do the same update before I saw your failure with the -F BIOS. I may be able to help though.
The label on the Flash IC reads:
A7528
INS320At boot-up in the top left corner:
A7528IMS V3.2 072208I may be able to give you a ROM dump if it's not against the Copyright rules on the forum.
Also can anyone find this M/B on the main site? All I can see are the
-F and -L(S) models. It's almost as if the plain G31M3 does not exist. I really need to update it because it only
maps 3GB of RAM even for 64 bit Linux and when I tried Win7 64 any process that updates the display would take
100% of a core (Vista 32 and Linux 64 seems fine with it).