Just to answer this question as simply as possible.
Gigabyte did at one time list the AMD Phenom II X4 940 for this board, They also had the bios required showing as F7.
However, If you check thier site now, They don't even show the 940 listed, But it was there at one time for sure, Personally I think its an oversight..
I know because I downloaded & updated the bios on my GA-MA770-DS3 ( revision 1) to F7.
I then fitted the AMD Phenom II X4 940, Thats was 3 months ago,Been running under Win Vista 64bit, I have had no problems what so ever, Its not overclocked as thats not my bag.
However the board runs sweet as a nut, Totally stable and very fast, Used almost exclusively for digital photo editing, Running Photoshop, Lightroom, Capture One 4, Photomatix, to name but a few.
Setup, AMD Phenom II X4 940, GA-MA770-DS3 ( revision 1), Asus 4350 ATI graphics, 8 gig of Corsair value 6400 ram, Creative Xfi Extreme music, 4 x Seagate 500gb sata 2 hdds, Corsair 620 watt PSU.