You say from a height so I'm assuming they are perching ok? If so, put a droppings board under the perch which you can easily remove for cleaning. It will catch the poop and therefore keep the floor below much cleaner.
Straw/hay is ok for the nests but keep it dry. Change it regularly. Shavings on the floor helps with the daily muck collected underfoot, but, the birds SHOULD be perching (perches are for sleeping, nests are for eggs) so use a droppings board.