Generally, .NET application should run the same on any Windows version with .NET framework installed. Most common problem that occur are related to the new User Account Control policy in Vista and 7, which prevents applications from accessing areas requiring elevated privileges (which would otherwise run fine under an Administrator account in XP).
For example, if your application is trying to write configuration settings to the Program Files folder (which is a really bad idea, btw), or is using some hardcoded disk or registry paths instead of environment variables provided in .NET classes, it may fail under Vista.
To quickly check if this is an UAC privilege problem, try to run the application as an administrator (have your client right click the .exe file and select Run as Administrator). If it works, then this suggests that you need to examine your code and update it to make sure you are only accessing allowed areas.
Check this link for more information: Making apps UAC aware.
As redsquare suggested, the best way to test your application in a variety of Windows systems is to run them in several virtual machines (MS Virtual PC or VMWare Player, both of them free for download).