At the moment, once you've checked out all the recent activity you might be interested in reading, the only way to clear the other items is to visit the threads anyway. A lot of forums (at least the older style ones like this) have a "mark all read" button on the index page you can use to essentially clear all of those. I'd like to request one!
(p.s. aleph, if you like the idea I can toss you code for it, I'm going to be writing it anyway for mine, just message me)