From what you say, this seems like an intermittent hardware problem. Likely your sound comes from some integrated sound chip soldered on the motherboard -- not a replaceable item. However if one of the PCI slots is not used on your system, you can add your own sound card. I usually end up doing that on most of my systems, as the integrated sound chips are usually crap.
When you add your own sound card, you will want to disable the integrated sound in the BIOS, to avoid conflicts.
Another thing I'd try would be booting off a bootable Linux CD, and see if it can find sound, but not everyone has a bootable Linux CD at hand, so that might not help you.
Is that E310 one of those BTX motherboards with the CPU near the front and the funky front vent:
The sound -might- be coming from an add-on board that's already in one of the two PCI slots (bottom right in the picture above, if that's your system.) In that case, try taking the card out and putting it back in -- could just be a loose connection between the card and the PCI connector.
No, you’re right the first time...it’s onboard.
It looks like a small (less than an inch) silver box. The plug-ins for the mic, speakers, etc. are all right there together.
I think I’m gonna go buy a sound card..
Chasaway