I will presume you're thinking of European poetry of Dante, Cavalcanti and Petrarca in particular as rhyme is found across the world, including pre-Columbian America and Ancient China.

Until a better answer comes around I'll discuss the academic opinions known to the modernist poet Ezra Pound way back in the beginning of the century.

The problem with writing a narrative is that we will have to connect dots of poetry different times and in different languages and these dots can be connected in different ways.

Well, the Florentine language poets explicitly harkened back to the troubadours of the Provence (Jaufré Rudel, Bernart de Ventandorn, Daniel Arnaut...). Their culture and language came to a downfall with the Cathar Crusade in the 13th century, but, presumably, their example has also been followed in Germany (Minnesang) and went, through France (Trouvères), to England, with the Anglo-Norman poets Maistre Wace, Richard the Lionheart and Tumas de Britanje. For some reason, they are rarely even mentionned as English poets, so that you get a gaping nothingness between the Middle English of a Geoffrey Chausser and the Beowulf.

Speaking of which: one of the academic opinions mentionned by Pound seeks the roots of a troubadour's rhyme in such Germanic staff rhymed poetry as Beowulf and the Ludwigslied. Here academics (mostly German ones) count the lines that rhyme and come to the conclusion that a fashion is arising to rhyme together with the middle ages. The French academics instead sought the beginning of troubadours in North Africa (which happened to be French at that time), by drawing parallels between Andalusian Arabic poetry (such as that of ibn Zaydun) and, through christian Spanish folk poetry (such as the cantigas de Santa Maria), to Southern France. Pound finds the French position more convincing than the German one but we still have no explicit credit given by any of the troubadours.