You can run .NET applications under Mono. But you may not be able to just take a compiled .NET exe that runs on Windows and expect it to run on Mono (although that was the original idea).
But unless you use any Windows specific stuff, you should be able to reuse most or all of your code, except that you need to make sure you test and debug under Mono/Linux too.
It is easier if you use Mono on Windows too, that way it becomes far easier to maintain it for both OSes.
Here's the link to Mono:
http://www.mono-project.com/Main_Page[
^]