You’d need to first upgrade to XNA 4.0. After that if you switch out XNA for MonoGame things should mostly work. It’s not that much work, I’d say.
Migration cheatsheet: https://web.archive.org/web/20110217153321/http://www.nelxon.com/blog/xna-3-1-to-xna-4-0-cheatsheet/
XNA to MG notes: http://www.monogame.net/documentation/?page=Porting_from_XNA
Note that the docs are written for the latest development version of MonoGame. The half pixel offset variable described in the second link does not exist in the 3.7.1 release (you won’t actually need it).
If you run into issues with effects (.fx files, which contain shaders), you can post back here for help. That’s where most of the friction is in porting because we use different tools to compile shaders and there are some issues with it. You’ll want to create a cross platform desktop project (AKA DesktopGL) so it runs on Linux.
Hope that helps!