Here's a fairly quick mash-up incorporating Yahoo Pipes and Flickr that I built a short while ago to address a simple need. Whilst Flickr provides plenty of feeds for differing data and such, one such feed that it doesn't offer is a single feed of all of your groups.
I set out to solve this using Yahoo Pipes, which was something that I wanted to play around with for a while. A few hours later and voila, Flickr - Get feed of all user Public Groups, a Yahoo Pipe that goes off to Flickr traversing your public groups, grabbing the feed URL and creating a single combined mega-feed. All that it requires to generate your specific group feed is your Flickr username!
Note that if you subscribe to a large amount of groups this is going to be a pretty hefty RSS feed so please be mindful of that.
Here's a screenshot of the elements involved in putting this together, click through to a larger version for a better view.