Here's my theory.
You need to run 12V power from the PC's power supply directly to the LCD.
The noise you are seeing is due to changing load on the PC's power supply (sounds like hard drive access), that you'll be hard pressed to filter out. That noise will travel up the VGA cable to your LCD.
If your LCD is powered from your PC PS, then your LCD is getting the same fluctuations in power to match the VGA signal, so you shouldn't see it.
Now if you power your LCD with an isolated power supply that floats, then that might work.
Also I'd imagine that a good ground from the PC power supply directly to the LCD's power gound should do the trick also.