Here is the sample game with video working in Linux:
I remember you that the video needs to be encoded with Theora.
I've used VLC to convert my videos.
I've precompiled MonoGame 3.5.1 with the changes to include it with the sample game, so you can just open the game project and run it.
The video used in the game is from the TheoraTestSuite. You can get more from here: https://wiki.xiph.org/TheoraTestsuite
If you prefer to compile MonoGame with this change for yourself:
1. Clone MonoGame 3.5.1 from the repository
2. Generate the solution with Protobuild
3. Open MonoGame.Framework.Linux.sln and under Media create a new directory Xiph
4. Add the 3 files included in MonoGameLinuxVideoTheora/src/Media/Xiph
5. Compile it and use it in your project
The credits should go to the FNA maintainer flibitijibibo and termoventilador for the other topics about this.
I tried it with MonoGame 3.6 but it only works if you comment #define VIDEOPLAYER_OPENGL in both Video.cs and VideoPlayer.cs so it uses the TheoraPlay software converter and avoids all the GL. mess... but it's a lot slower...
Maybe someone more experienced than me can give some help with the missing OpenGL functions...
Hope this helps you in some way