Mono is an open source .NET framework that's better than .NET in a lot of ways. A lot of .NET stuff on Windows actually uses the Mono runtime rather than MS's .NET ones. So there's some chance that it'll 'just work' on Linux.

Copy the program to a Linux machine, make sure Mono is installed then chmod +x the .exe then try running it from a terminal. You may get missing .dll errors, if so just copy those .dlls across, putting them in the same directory as the .exe and try again. This can often work.

Otherwise, porting from .NET to Mono is very easy, so it might be worth nudging whoever makes this and nicely asking if they can do a Linux build.

Failing that, yeah, Wine should work fine as Ant says.