I have a 3 year old PC that has started freezing randomly and I'd appreciate help figuring out how to fix it.

The problem started when I moved the PC to a different location in the house. The PC freezes only when it is doing something. That is, if it is left sitting at the desktop with no applications open and with the screen saver off, then it is fine. It is also fine when left sitting in safe mode. But if it is doing something - like installing Windows, web browsing, or even just running the screen saver with no applications open - then it freezes at seemingly random times.

When it freezes, the screen is unresponsive, the hard drive is inactive, and the mouse's light has switched off.

Things I have tried, with no effect:
- Swapped to a different hard drive.
- A fresh install of Windows 7 64-bit on the different hard drive. Minimal software installed, though with 100+ Windows updates.
- Swapped to a different video card.
- Removed and reinserted the RAM.
- Disconnected the network cable.
- Changed to a different keyboard and mouse (first set was wireless, second set is USB).
- Run the Windows memory test, which was OK.
- Run burn-in software that tests the CPU and RAM at 100% for 8 hours (with screen saver off), with no errors.

Any suggestions about what the problem might be and how to fix it?