You don't need to make your game open source.
If you plan to give the game away for free, why wouldn't you do it properly? I think that it's a great initiative to make your game open source, rather than freeware. It benefits both fans and developers, who can help patch the game or build new things from it. You still retain the right to do anything you want with it; including creating future, non-free versions of it.
As for the actual question: If your game runs on Linux, you will be able to distribute it as an official package for different Linux distributions. There are already many games available on the official Debian servers (both good and very bad games). If your game gets accepted there (which shouldn't be that hard) it will automatically be available in the Ubuntu Software Center (USF). Once there it is one-click-away for everyone using Ubuntu/Kubuntu/LinuxMint/etc.. If your game gets good ratings in the USF it will likely become promoted as "recommended software", which will expose it to millions of users.edit:
A tip for the open source release: upload it to github. That way it will be super-easy for others to submit patches, report problems and create forks. It will also be a nice feather in you cap.