Note: Not all of these apps are strictly Linux only; at least one has Windows and Unix versions as well.
This is a self-contained news feed aggregator which is available for a number of Linux and Unix distros. I know that you can import an XML file of RSS feeds but it doesn't appear to have any integration with external services.
Here's one which appears to have been abandoned since 2010 (no updates, at least) so it's not clear to me whether this app will operate independent of Google Reader.
This is the app that caught my eye yesterday (thanks, Doug!).
Unlike other news reader apps Liferea can be used independently or it can be used with does require an external service like Google Reader. Luckily this app can also integrate with Tiny Tiny RSS, making it a good option for the Linux user who wants to use a single account on multiple devices.
This app is currently under development and is already set to survive the Readerpocalypse. It has different versions for Linux, Unix, and Windows.