Follow up questions: Did humans have better memories "back in the day"? My parents generation was forced to memorise long passages during school (Shakespeare etc)... are the techniques they used similar to the ones the ancients would have used? Are the script memorisation techniques used by actors today similar to what was practised historically?
Memory is pliable. Training in mnemonic techniques can be used to improve memory. The military academies require multiple things to be memorized at all times to enhance memory improvement.
Epic poems weren't recited word-for-word every time, at least not in our sense of the word. A poet would have a bag of tricks - epithets, phrases, and stock paragraphs - that he could use to stick with the meter and keep the story going. The technique has often been compared to jazz. A jazz trombonist can draw from a number of riffs and tricks to play along with a tune. Even if he's played it fifty times, it might come out differently every time, but it's the same song. Same with any blues standard. Different every time, but the same every time. You eventually get to where you got to be.
Milman Parry (and Albert Lord) studied the bards and oral tradition in Yugoslavia, which he then compared to Homeric recitation. When he'd ask a bard to repeat exactly the performance from the night before, they'd riff as always. When Parry noted the differences, they'd insist it was the same tale. Because it's the meter and the main elements of the tale that matter. The exact words don't. The concept of an absolute fixed rendition would be bizarre to a bard.
The Parry and Lord stuff makes for some interesting reading.
Some ancients used the method of loci. It's first mentioned in Rhetorica ad Herennium, and again in Cicero's De Oratore, and Quintilian's Institutio Oratoria. Basically, you use mnemonic devices to associate whatever you want to remember in a particular order according to an already familiar mental "journey."
I took a class on it once. It's really easy to learn and it's surprisingly effective.
Hi, seems to me I recall seeing this question ~3 years ago in /r/AskAnthropology, so you may want to check there (and maybe x-post too), but meanwhile, I have found a few posts here that may be interesting
How do historians feel about drawing conclusions from oral traditions? how accurate are they deemed, professionally speaking? - some mention of techniques in /u/earthvexing_dewberry's comment