I'm not sure this is the right sub to ask, because it's mostly a practical problem. In oral traditions, like poetry told by Bards through even modern Jazz and Comedy (or even rap, which often isn't written down), you don't memorize every line, note, word, etc. You remember the major plot points, then fill in the details.

For example, take a childhood story like the tortoise and the hair, or even a biblical story like Moses leading the Jews out of Egypt. You can remember the major plot points, right? If you were asked to recount the story, the art of being a bard is to fill in those details in a clever way. Someone else can remember those points, but change the details later, and continue on through generations. That's how you end up with myths and legends.

I am currently reading the Barnes and Noble Classics edition of The Odyssey. In the analysis that preludes the story they talk about how the bard would remember the major plot points and key phrases like "earth-shaking Poseidon" and would have characters repeat word for word what they were previously told in order to keep the information simple and easy to remember.

They also talk about how bards would change it up based on their audience adding different descriptions that would be found to be more interesting to the audience currently at hand.

The B&N Classic editions of both The Iliad and the Odyssey have some great analysis dissecting the poem and you should pick them up if you can

Your question reminded me of the Manas epic told by Kyrgyz people. It is a huge poem that is recited line by line like a chant. Just one of what I'm sure are many examples of how epic stories were passed down before writing. Memorizing would be the only way to do it, but having the tale in the form of a song or chant probably made it much easier to memorize. http://youtu.be/ghUmtdWhU5o