Hello,
I own a Gigabyte HD5750 Silent Cell, which isn't really supported (I'm stuck with 1400x1050 without any acceleration, video etc.). This is due to the fact that it appears to be a non-"reference design". In my case the connectors (VGA, HDMI and DVI-I and DVI-D) do not correspond to those of supported cards (usually DP, DVI-I, HDMI, not necessarily in this order). The next higher model, the Gigabyte HD5770 Silent Cell, has the "correct" connectors, so I believe it would work. Its "sister" card, the HD5770 "Batmobile", seems to work pretty well (it even got an "appreciation" thread!).
My advice for you: Look for a supported card among the various success posts in this forum and see if you can find a fanless version from the same manufacturer. Check the connectors, and if they correspond to those of the supported version, you may have a winner... ;-)
HTH,
brazza