Computer won't show up on any monitor

When you say everything is booting up properly it might not be case, not because you hear the fans spinning means that the system is booting. One way to tell is to look at your hard drive activity LED if it's flashing repeatedly and you can hear the drive working hard it could mean that the system is booting indeed because the OS is loading from the HD. Then, you could try to access the system remotely through the network, or at least try to ping the hostname or IP address then it will tell you that your OS has loaded and is alive and that it's only a display issue. But as I said even though some lights are coming on and some fans are spinning it doesn't mean that the full boot process is going through, for example if the motherboard or power supply have problems they could just send enough power or signal for the fans to spin, but fail to launch other vital components like the CPU or video card or some chipsets, etc...

Try to boot with minimal components, remove the video card and try onboard video. You should also remove everything else, all peripherals, all expansion cards, even disconnect the CD-ROM and the hard drive (you don't need the hard drive to get to the BIOS, from what I understand you're not even able to display the BIOS, right?). You can even disconnect the mouse, because to access the BIOS all you need really is only onboard video and keyboard, that's all. The BIOS is the most simple thing to display on the screen because it doesn't involve any drivers, but if you can't even display that with onboard video, I'm afraid some major components could be defective like the CPU, motherboard, power supply, etc...

Thanks for all your suggestions! I tried them all, but unfortunately none of them worked which means my ol' desktop is a gonner. I guess I'll just have to slip into a deep dark depression and begin the long search for another.