Sounds a lot like using a coffee sock. I haven't tried it, but looks easy and effective:
http://hubpages.com/hub/How-to-use-a-coffee-sock-A-cheap-way-to-make-coffee
Thanks for that link. Although I've never used a sock for this purpose, the concept is the same. The trick is in finding a strainer fine enough, otherwise you end up with quite a bit of sludge. I generally take some time to sip my coffee, so by the time I'm done, the sediment usually settles at the bottom of the cup.