The traditional way to have a C# MonoGame Project compile both under Linux and Windows is to use a tool named Protobuild. With the last version (3.6), the MonoGame OpenGL template for a cross platform game includes references to the libraries in the three main formats: DLL for windows, DyLib for mac, and so for Linux. This seemed like a good chance for making things work.
And it mostly did. Change to the path where the .csproj file resides, and run xbuild. The bin folder should be created, and inside there, a path accordingly with your project configuration. For example, bin/DesktopGL/AnyCPU/Debug. Then, go there, and just run your game with
(future work: check how to make things work outside the output dir)