Drs. Foster and Smith, from whom I order far too much fish-related equipment, use biodegradable packing peanuts in their shipments. This makes getting rid of the peanuts easy, if you don’t want to reuse them: just put them in in some water and they disappear. They are even supposed to be compost safe, being made of some sort of starch material. What you have to be careful with, if you are playing with an aquarium and unpacking your newly-arrived toys, is that this doesn’t happen:
Remember, if things degrade in water, and your hands are wet, the peanuts will decompose and stick to your hands. Of course, it isn’t a big deal…just run your hand under some water and all the packing-peanuts wash off/disappear.