So yesterday my computer started making weird sounds like 'thuds' and 'blips'. I thought little of it, probably just some feedback from the microphone or something and it was done. Then later on that day it randomly restarted on me. So I started becoming slightly concerned especially when the sounds were now more frequent. Its been over a day now and the sounds are very frequent (heard like 6 since I started writing this) and my computer has randomly restarted twice today (both time within 5 minutes of each other).

Normally I'm able to solve my own computer problems but this one baffles me, I can't even imagine what might be causing it and I unplugged my microphone so I know its not that.

The sound

Hm, when a computer randomly restarts it might be an overheat issue caused by a malfunctioning fan.
It could also be the power supply fan, or the power supply itself that is dying.

Clean out the computer. This happened to my laptop and we cleaned all the dust that had accumulated in it and it ran fine after that.

I just cleaned it out a week and a half ago. Its not overheating or a virus, I've ruled those two out (just moments ago ><). My roommate thinks it might be a power supply issue but my power supply seems to be in very good condition.

Blah ... I fixed the problem but to do this I had to swap out my processor for an older one. Not a huge deal 2.0 to 1.8 but a downgrade is a downgrade. Side thought, anyone know of any sites that let you build to order a computer (without forcing you to get certain things like ram, OS, HD, etc) and let you pay for it monthly.

Are you using Windows XP or Vista? They've replaced BSoD with an automatic restart, so that could be it. I've heard there's an option somewhere to turn that off...

Where are these sounds coming from? Are they coming from the speakers, or from the computer itself?

They're coming from the speakers (or headphones if those were plugged in). I am using XP right now and I still do get a BSOD rarely from XP but these weren't BSOD incidents.