While quality native Linux games are rather in short supply, those that do end up coming out of the professional game studios end up being first-person shooters, just look at Doom 3, Quake 4, Enemy Territory: Quake Wars, Cold War, Unreal Tournament 2004, etc. Even on the open-source side there are many first-person shooters from Nexuiz to Warsow to many others. If though you have been trying to find a new non-FPS game that has native Linux support, there is a new one emerging and that is QuantZ.
QuantZ comes out of a Canadian game studio known as Gamerizon and this is their lone title currently. QuantZ is self-described as "a unique puzzle action game", but fear not it's far from being an ordinary puzzle game. This game is OpenGL driven and appears quite polished from initial testing. Below is a trailer for Gamerizon's QuantZ.
The Windows version of this game is currently selling for $13 USD, but Gamerizon is currently offering up their current Linux build of QuantZ for free before its officially released. They are offering the (closed-source) Linux version in RPM, Deb, and .tar.gz forms. Gamerizon announced this information on the social bookmarking site Reddit.