XNA is a new set of tools for assisting in game design, development and mangement (see the XNA Wikipedia page).
One of the components is the "XNA Framework" a set of .NET APIs for actually building the applications. It does not expose DirectX which makes it simpler to be ported to other platforms.
Rob Loach has started a project to do an open source implementation using the TaoFramework (Tao provides bindings for all things OpenGL and a few other media libraries).
Rob's implementation is being developed here. Currently most of the development is happening on Windows, but the project on SVN has support for building this on Linux with Mono as well.