so does it happen if you stay idle on the desktop?
This happened with my PC once and the problem was related to the hard disk. Considering that chkdsk reported errors, you might want to backup your data.
No harm in updating the BIOS if its a straight forward procedure with your motherboard.
It could be the CPU, but since it works fine during normal use, it doesn't add up. The only way to confirm is to run the PC with a different CPU.
It could be your older HDD. You can try to disable it using the BIOS. What softwares do you have running in the background? Most maintenance software will do their processing when the CPU is idle so that it doesn't slow down your PC during normal use. If its trying to read from the disk constantly and your HDD is faulty, it could be the problem.
If you are getting the BSOD, it would certainly help if you post the minidump file.
I don't think he's getting a BSOD. But that certainly seems like a plausible issue (the old hard disk causing the issue). The indexing service, for example, runs in the background. With more cores and threads available with your new CPU, this may have magnified the issue (or possibly create a deadlock that couldn't be reached with your older CPU).
As AbBh said, try disabling your faulty HDD from BIOS startup and running with that setup for a few days. If that doesn't solve the issue, at least you have eliminated one of the potential culprits.
And you should also run memtest86+ at some point just to make sure your memory checks out.
Though obviously if they don't have a CPU for that board in stock, then it's a bit trickier to use the SoGA.
It sounds like you're booting from you old HDD which means if you want them to remove it you'll probably have to reinstall Windows on the new HDD.
The boot loader is on your old hard drive I think.
So you had a vista/xp installed on your old hard drive, then got a new harddrive, and installed windows 7 on that in a dual boot?
I think you'll need to make the windows 7 boot the active partition, or something along those lines