I recently went through approximately 90 of my Windows games to see which still worked on my new Windows 7 64-bit machine. Most worked fine, but there were many that had problems. There were some common problems you might encounter:
Games using 8-bit colour will display with random colours. This affects a lot old Windows games, like Starcraft, Master of Orion 2 and Age of Empires and apparently happens on both the 32-bit and 64-bit versions of Windows 7. There's a simple, if inelegant, generic solution to the problem: Kill Explorer in the Task Manager after launching the game, and then restart it when you're finished. It's also possible to fix the problem with a registery hack specific to each game that has this issue. The registry hack for current versions of Starcraft isn't too hard to find on the WWW.
Games that required some random hack to get working. This covers a large range of issues, but ultimately the solution was to search the WWW to find the fix. You shouldn't need to download anything (although the using the lastest offical patches is often a good idea), but you might need to edit the registry or an .ini file.
Note, that enabling compatiblity mode options (the ones you get right by right clicking and selecting Properties) weren't necessary to get any of my games working except for one. In that one case I just needed to select Run as Admininstrator. While turning on various compatibilty options like is this a common solution you'll posted on WWW forums when anyone has just about any problem with Windows, it's very rarely going to fix the problem. Windows has large a backwards compatibility database that will automatically enable a much wider range of compatiblity shims that allow many old Windows games to just work. If this targeted and more precise automatic functionality doesn't work, then the manual options aren't likely to work either, unless you've got a really obscure game broken the right way.
Games that required using software rendering. A few of my old games (eg. X-Wing Alliance, Mechwarrior 3) have graphic corruption with hardware rendering turned on, but worked fine with software rendering. This is probably a problem specific to my ATI Radeon 5770, not Windows 7. You might not encounter it with a Nvidia card or an older ATI card. Of course you may encounter rendering issues with a different set of games instead.
Games with broken copy protection. This actually only affected a couple of my games. The 64-bit versions of Windows have special support for a lot of the 32-bit drivers that the various copy protection schemes use. In one case I was able use a No-CD crack to solve the problem, in another, Beyond Good and Evil, there doesn't seem to be a proper crack available that will get it working on 64-bit versions of Windows.
Games that require support for 16-bit applications. A couple of my old games, Civilization II and the original Colonization for Windows are 16-bit Windows games, and so don't work on the 64-bit versions of Windows. Since these are Windows games, not MS-DOS games, Dosbox isn't a solution. You'd have to either use some virtual machine software and an appropriate version of Windows, or another machine to play any 16-bit Windows games you might have.
Note that many, many, old Windows games have 16-bit installers, despite otherwise being fully 32-bit games. This wasn't a problem for me, since I had most of my old games still installed, and because Windows also includes special support for running these 16-bit installers. If it's a problem for you might need to install the game on another machine (or maybe a virtual machine) and then copy it over. You might also need to copy over some registry entires, which is tricky enough, but harder in this case because some of them will need to be copied to a different place on 64-bit Windows.
The were only a few a games that ulimitately I couldn't find any good solution for. I've already metioned Beyond Good and Evil, I also couldn't get Jedi Knight and it's expansion pack Mysteries of the Sith running without crashing. Microsoft Flight Simulator for Windows 95 also didn't work right for me, but I didn't look into that very closely. I only managed to get Jedi Knight 2: Jedi Outcast working by coming up with my own patch to fix a bug in the game.
Oh, like DaveSimmons suggested, I didn't install any of these games to the default Program Files or Program Files (x86) directory. I installed them to seperate exFAT partition where I didn't have to worry about NTFS permissions causing problems. A FAT32 partion should work just as well.