When I turn on my Logitech Z4 Speaker system, all I get is a buzz/hum. It comes though the sub woofer, then when I plug in the two speakers, it comes through them. Has this happened to anyone else?

It's a problem of grounding means your speaker input is not getting proper grounding through cable. when i was setting up my sound system in my room i got same problem i tried so many ways to remove humming of speaker and sub, i also bought a cable extender for this issue but problem was not solved. so i bought a good AUX cable and connected with my workstation and Humming of sound system gone. another solution is try to put your system away from Electromagnetic equipments i-e router, cell phones, transformer of speaker system etc..

It may be an earth loop, (ground loop). If the mains for your PC is plugged in some distance away from where the Logitech is plugged into the mains. Try to plug them both in side by side into the mains (or even use piggy-back mains plugs if available).

This sounds like EMI, I would try to move the speakers away from anything and also the wall. See if that helps.
this also might help as well.
http://www.ehow.com/how_4856392_troubleshoot-buzzing-sound-stereo-speakers.html

Done forget to vote for me :D

Make sure your area is clear of any other electronics - especially ones using radio (like cell phones). there could be an issue with interference and you may need shielded cables if you cant get the offending devices away from the speakers.

Lower the volume and might need higher quality sound card that will not pass through computer parts interference.