You could try one of these:
- Build Mac versions on a Mac and Linux versions on Linux. Ugh.
- Use .NET Core (template here) to build self-contained releases that run on Windows, Mac, or Linux without requiring .NET Framework. This is pretty cool, but was honestly a nightmare for me to setup and I cannot describe it all here.
- Port your game from MonoGame to FNA. (Shouldn't be difficult.)
- Use MonoKickstart to package Mac and Linux runtimes with your game.