I think Jenny's suggestions are good ones. This is really starting to sound like a software issue more than hardware--probably having something to do with whatever was causing problems under Vista.
All the same, I think we should try to rule out some of the most likely hardware problems as well. The most common hardware-based cause of these sorts of issues is probably overheating. The easiest way I know of to check for overheating is to download a copy of the free version of CPUID HWMonitor
. Just click the "1.17 setup" link under "Download latest release", then download, install, and run the program. Note that Windows may ask for permission to run the software: make sure you give it permission so it can find what it needs. Also, it's possible the software will ask if you want to also install bundled software like Google toolbar. Make sure you DON'T install whatever it bundles (well, unless you really want to). Otherwise you'll just have to uninstall it to get rid of it.
Once HWMonitor starts up, you should get a list of temperature readouts for a bunch of hardware on your computer. You'll want to pay attention to a few rows in particular: the rows corresponding to your CPU (probably listed as an Intel Core 2 Quad, or similar) and your video card (probably listed as GeForce 8600GT) are good to start with. They'll both list 3 temperatures: Value (which is the current reported temperature), Min and Max. You may need to click the plus sign to the left of both rows to see the listed temperatures.
In a healthy computer at idle, the CPU should be between 25 and 50 degrees or so, and the video card should be hovering somewhere between about 35 and 60 degrees Celsius, depending on the card and the case it's sitting in. If the temperature of the video card is already north of 75 degrees (Celsius), you may have an overheating problem.
Also take a look at the other temperatures reported and make sure nothing else looks out of line. You may see a couple of erroneous temperatures reported, (my motherboard reports an AUXTIN temperature of 9 degrees Celsius,) but mostly keep an eye out for anything exceeding 70 degrees Celsius (or 75-80 degrees for a video card.
Once you've taken a look at your idle temperatures (and written down some of the important ones), leave HWMonitor running in the background, and start up one of the games that gives you trouble and play it for a few minutes. Then alt-tab back to HWMonitor and check your temperatures again.
Games stress components, so you should expect to see some temperature increases. However, you should be keeping an eye out for any dramatic temperature spikes. Video cards shouldn't get any hotter than about 80-85 degrees under load, and ideally they should be closer to 60-70 degrees. CPUs shouldn't go above about 65 or 70.
Note that it's handy to keep track of the "Max" column, since that will tell you the maximum temperature reported so far, even if you don't happen to be observing the computer when it happens.
If you're seeing alarmingly high temperatures anywhere, quit the game for 5 or 10 minutes and watch to see if the temperature drops, and how long it takes to regularize. Otherwise, keep playing and checking your temperatures from time to time. Either way, if you report your observations here, we can tell you whether the temperatures you see are anything to worry about.
If you don't want to write too much stuff down and you're feeling particularly adventurous, then for each step where you check HWMonitor, you can put HWMonitor in fullscreen (so you can see everything,) printscreen the results, paste them into your favorite image editing software, and save the file somewhere so you have a reference. If you save a printscreen from the computer at idle, and one after you've been playing for 5-10 minutes, that should give most of the data you'll need.
I'd also suggest running a chkdsk /r on your OS drive. If you need instructions for how to do so, ask. Note you'll have to leave the computer running on its own for quite a few hours for it to complete a chkdsk, so I suggest you let it chug overnight while you sleep.
It also wouldn't hurt to chkdsk any other hard drives in your system.
While it's not likely this is a hardware problem, I do suggest you try both diagnostic procedures above to rule out a) temperature issues and b) impending hard drive failure. If this IS a hardware issue, you should know about it now rather than later.