All ancient poetry that I can think of (Hebrew, Latin, Greek, Sanskrit, Germanic, Chinese) did not rhyme.
The earliest rhymed poetry I’m aware of was Arabian. The earliest extant Arabic poems date to the pre-Islamic era, circa the 5th century AD. All Arabic poems rhymed from their earliest appearance. Not only that, even the prose in Classical Arabic usually was rhymed. Modern hip-hop rap lyrics are remarkably close in style to old Arabic rhymed prose.
Rhyme first appeared in French and other European poetry sometime after the year 1000, correct me if I’m wrong. The Middle English text of Sir Gawaine and the Greene Knight makes reference to the recent vogue of rhyming poetry and the abandonment of earlier alliteration in English poetry. That period in the Middle Ages (c. 1000-1300) is when Arabic/Islamic culture filtered into Europe via Islamic Spain and Sicily, and the Crusades. That Arab literature influenced European literature in this period has been demonstrated by many scholars. (Cites? See, for example, the studies of Dante by Miguel Asín Palacios like Islam and the Divine Comedy; Love in the Western World by Denis de Rougement; and España, eslabón entre la Cristiandad y el Islam by Ramón Menéndez Pidal).
Beginning in the 10th century, Persian poetry reappeared after Persian literature had been through a hiatus following the Arab/Islamic conquest of Persia. It now rhymed. I don’t think ancient Avestan poetry rhymed. Likewise, Hindi poetry in the late medieval period began to rhyme under the influence of Persian poetry.
Countries that did not have Arab cultural influence (like China, Thailand, Japan, American Indians) never developed rhyming poetry. Countries that did have medieval cultural exchange with Arabs did start to rhyme, beginning from that time.
So the implication of all of this is that rhyme originated in Arabia and its occurrence in all other languages from Europe to India is traceable to Arabic influence. I’m so glad I discovered this I could burst — Remember, folks, you read it here first.