Aristophanes The Birds has some scenes where people imitate the sounds of birds.

Here's an example from lines 227-262:

Epopopoi popoi popopopoi popoi, here, here, quick, quick, quick, my comrades in the air; [230] all you who pillage the fertile lands of the husbandmen, the numberless tribes who gather and devour the barley seeds, the swift flying race that sings so sweetly. And you whose gentle twitter [235] resounds through the fields with the little cry of tiotiotiotiotiotiotiotio; and you who hop about the branches of the ivy in the gardens; [240] the mountain birds, who feed on the wild olive-berries or the arbutus, hurry to come at my call, trioto, trioto, totobrix; you also, who snap up the sharp-stinging [245] gnats in the marshy vales, and you who dwell in the fine plain of Marathon, all damp with dew, and you, the francolin with speckled wings; [250] you too, the halcyons, who flit over the swelling waves of the sea, come hither to hear the tidings; let all the tribes of long-necked birds assemble here; [255] know that a clever old man has come to us, bringing an entirely new idea and proposing great reforms. Let all come to the debate here, here, here, here. [260] Torotorotorotorotix, kikkabau, kikkabau, torotorotorolililix.

There are other scenes with different birds that go "popopopopo" and "tiotiotiotinx."

In many cases, the words used for animal sounds were also words for generic groaning, bellowing, or murmuring. Take the Latin word "mugio," which often refers to the noise that cows make but could also refer to the sound timbers make when they bend as on a ship. This is cognate with Greek "μύζω" which just means "make a μύ sound" and could refer to crying, moaning, or growling and was used by Aristophanes to represent a dolphin sound.

Ohhhh man, this is fun! We have lots of Latin-language literature that mentions animals, so we actually have a pretty decent vocabulary of animal sounds. For instance, Catullus writes a poem about the death of his girlfriend's cherished pet bird, who always used to chirp on her lap (Catullus III). Vergil's Georgics is an entire freakin' book about animal husbandry, so there are plenty of animal sounds in there. As /u/CorneliusNepos mentioned, a lot of animal sounds are also words for human sounds; a wolf can howl (ululo), but so can humans (see Aeneid IV and the ululating wood-nymphs).

A lot of them are actually pretty similar to the English versions, or at least make sense as onomatopoeias. Here are a few off the top of my head, verified with Lewis&Short:

• If I were a sheep, I'd balo = bleat. (I'm giving the first person present active singular forms for all of these verbs, just for ease and consistency.)
• If I were a cow, I'd mugio, mugire = moo. (Screw it, I'll do the first two principal parts.)
• If I were a dog, I'd latro, latrare = bark. (This one isn't especially onomatopoeic, but there are other words that are more so, like baubo.)
• Bird = pipio, pipiare = tweet/chirp/peep.
• Snake = sibilo, sibilare = hiss. (Think of the English word sibilant.)
• Horse = hinnio, hinniare = neigh/whinny. (Hinnio even sounds like whinny!)