If you're lucky, the cushions come off and the covers on them zip off. Then you can wash the covers and give the open-cell foam inside a bath, or even replace it if needed.
Less lucky, and you get my guesswork. I've heard great things about cleaning carpet by pouring on fairly large amounts of water, then sucking it out with a shopvac. Depending on the material, doing this and putting it somewhere it can dry well might do the trick.
Maybe you need to invest in that granny couch shrink-wrap.