There are a few classic instrumentals that incorporate animal sounds: Fox Chase, Cluck Old Hen, Chicken Reel, Cumberland Mountain Fox Chase and Listen to the Mocking Bird come to mind.

There are a bunch of songs with such imitations: I Had a Rooster, Old MacDonald Had a Farm, Little Pigs Lie in the Best of Straw, There was an Old Lady Who LOved a Swine, Susanna's a Funny Old (or Funicle) Man, Old Blue... My kids' favorite was Fod.