The solution to this issue has been posted here:
viewtopic.php?f=14&t=83652
In short, in VLC's preferences, under the "Video" settings, change the output to "Direct3D Video Output", or in older versions, "DirectX 3D Video Output". And if that doesn't work, try each other output setting until you find one that works. "Windows GDI Video Output" is what did it for me. And stop (don't pause) the video while changing the setting, or it won't take effect.