Ever since I moved to Linux, I've been looking for a RSS reader that I like as much as Great News. I have finally found one that comes very close in Brief. Brief is a Firefox extension that adds a very simple, Great news-like feed reader to Firefox. It even stores the feeds locally for as long as you like, and just like Great News "as long as you like" can differ with each feed. It does not have some of the whiz-bang features of Great News, but if has most of the ones I regularly used.
From the Brief web description on addons.mozilla.org: