A 3-letter word for a type of tropical American cuckoo is an ani.
The ani is a bird of medium size, shaggy and black, but with bronze overtones. They have a thick bill and an often bobbed long tail which is held beneath the body.
The ani have an interesting breeding method, they are communal nest builders. In other words, several pairs of birds will construct a single nest into which the females will lay their eggs, in layers, separated by grass or leaves; there may be as many as 30 eggs in a single nest.
One of the collective nouns for a group of anis is a silliness of anis. Two others are a cooch, and an orphanage.