I hit the boing and the rope perches with Poop Off, scrub with a soft brush to get rid of any stuck on bombs, then give them a good steaming. I try to position them in the cage so that they are not in a heavy poop zone (like under where he sleeps at night).
I do put them back slightly damp. They dry pretty quickly after the hot steam. I also take them outside and swing them around over my head (I must look like a complete lunatic to any watching neighbors) to try and get rid of any extra water in them. They really feel dry to the touch after that, although I'm sure there's still some moisture inside the rope.