One thing you could try is deleting the file [My Documents]\Bioware\Mass Effect\Published\CookedPC\LocalShaderCache*.
You could also try lowering the GPU memory clock to see if it helps.
I know that this is really late, but I will put it here in case someone else stumbles over it. It seems to be caused by Mass Effect using 3D Now!, which your CPU and newer ones don't support. To fix it, although it isn't perfect, go to Documents\BioWare\Mass Effect\Config\
Open BioInput.ini and under [Engine.Console] add a line ConsoleKey=Tilde
Save the file
Open Mass Effect and in the affected areas, Noveria and Ilos
press the tilde key
input: viewmode unlit
to undo viewmode unlit use the command viewmode lit
Areas where the black blobs go away:
Noveria after you take the tram to the barracks and hot labs area.
Ilos at the end of the level.
If the black blobs on your characters persist after these areas, either equip a different armor and reequip the armor you want, or save and reload the game.
The reason this isn't perfect is because it takes away the lighting and shadow effects, but I find it preferable to the blocky black blobs. Also I don't know if it is just me, but in areas where the black blobs are, the autosave function doesn't work, so be sure to make manual saves in these areas.
I hope this helps.