1) RMB the part in the assembly, and go to properties of the part.
2) You will see a Color button. Click that button.
3) You will get another Dialog box. Click Remove color.
4) Once you Click "Remove" the color goes black, but that doesn't mean that the part is going to use the color black. Click OK
5) Click OK again.
What this does is, it turns off the assembly color and now you should be using the Part color.
SW03 and future Versions:
To change the color of a part at the assembly stage there are now two ways of making this change.
RMB the part in the assembly and either Click "Properties" or "Appearance"
Properties =
Click the "Color" button and select the color you want for the part at the assembly stage.
Appearance =
Click "Color" in the flyout menu. In the Color Properties in the Property Manager you can select the Color of the part your seeking.