The Short: Get a better cable - the monitor cable, I mean.
The Long: This is pretty much an EMI issue far as I can tell, so lomac is kind of right though the reason is a little more involved.
The video subsystem uses very high frequency to communicate. Due to this the system impedance needs to be very low to prevent noise pickup. A second truth is that high frequency signals generally travel on the surface of the conductor - this is known as skin effect.
A badly designed cable will have higher impedance on its terminations and possibly along the cable length as well, causing a (very small) voltage drop. This drop will then be modulated by surrounding electrical fields. The shielding is critical for a second reason - high frequency can be easily modulated by low-frequency EMI (very common in power cables, which are always unshielded for safety reasons) from neighboring cables or circuitry.
A poor quality shield (<90% coverage may be considered poor for this application) will be unable to prevent the modulation. The problem is compounded by the skin effect (you won't see similar phenomenon in most decently-designed audio circuitry as the frequencies are much lower). That is most likely the reason for the interference. Has nothing to do with your monitor, IMO.
DVI is generally less susceptible to such interference, but is not immune. Cables bundled with monitors are generally quite poor quality, and a problem like this indicates the need for a new cable.
