I am pretty sure I found the problem. Either the new hard drive is bad (I bought a new HDD and SSD for this machine and transferred over one SSD and two HDDs from the old machine) or the SATA 2 port is bad.

I was able to reliably lock the machine up just by waking it up from sleep mode and selecting the Start button. I noticed that while transferring data from the new hard drive, it wasn't waking up after sleeping (drive not responding errors). Simply disconnecting it seems to have solved the problem, and I've been running for 3 days without a lockup.

Now to figure out if it was the port or the drive...

Thanks for all the help, folks. You never cease to impress the hell out of me and this is another thread to bookmark for future reference.