Do not install those games (especially old games) in the 'Program Files' folder. Vista and Windows 7 has put restrictions on access to the 'Program Files' folder which may cause incompatibilities with older games. I personally install old games in the 'C:\Games' folder. Also, try running the game as an administrator (right-click the game's .EXE file to do this). If it fails, run as administrator and try running in Windows XP compatibility mode, if that fails, try Windows 98 compatibility mode.
Also, make sure you are using the latest version of the game you are playing as newer updates might include performance and compatibility improvements on newer OSes asides from game bug fixes.
I search for game updates at this website: http://patches-scrolls.de/
I have both a computer with Windows XP 32 bit installed and another one with Windows 7 64 bit. I play lots of 90's-early 2000's games in both without any problems. In fact I haven't experienced an old Windows/DOS game not running properly.
More info on compatibility modes for Windows 7:http://www.sevenforums.com/tutorials/316-compatibility-mode.html
http://duckduckgo.com/?q=%22Windows+7%22+compatibility+mode&v=
Run a program as an administrator on Windows 7:
http://www.sevenforums.com/tutorials/11841-run-administrator.html