After a quick search for RSS aggregators, I found Google Reader to do exactly what I want. It saves so much of my time monitoring RSS feeds from a central location, and being web-based means I can access it from anywhere. In case you are wondering, I don't work for Google!
Update: Since publishing this, I've had a few people tell me how much Google Reader sucks compared to the alternatives out there. Greg Black recommended Bloglines to me, and after giving it a go I must say that it's a much better solution. Looks like I'll stick with this one for a while.