Here are 5 of the most popular, standalone feed reader applications for Linux. I didn't include feed readers which come with applications like Firefox, Opera or Thunderbird for example, but these can also be considered a viable alternative to the ones below.
Akregator is the default KDE feed reader, offering a highly configurable application, support for RSS and Atom feeds, very good integration with KDE, tabs for the internal browser and feed management. Akregator comes with system tray integration, feeds notifications, interval fetching, ability to keep feeds, browser cache, font customisability for the internal browser, tabs, external browser can be changed from the default. Akregator is included in the kdepim package which comes with KDE4.