I am a big fan of both medieval and Anglo-Saxon literature for this very reason! Up until the invention of the printing press, Western European culture was oral rather than written. This means that people would have to memorize literature and poetry in order to share it aloud with others. What you'll find is that medieval and Anglo-Saxon literature is very easy to memorize for this very reason.

I recently learned something really cool about the oral tradition. When writing first came about in what had previously been an oral culture, scribes did not include spaces between their words. This was because written words were meant to mimic oral speech. When you're speaking, you don't pause in between each word, or even every sentence, so it would make sense to record words without spaces. What did this mean for reading? Well, since it was written to mimic oral speech, writing was also meant to be read aloud. It wasn't until the invention of the word space that reading started to take place in silence and seclusion. Whereas before, reading was a social activity, it became a solitary activity done in private after the word space came about.

I think trying to memorize a new poem every day is a great idea! I would recommend trying out some medieval English works because they really lend themselves to memorization and oral recitation. If you're truly ambitious, you could even memorize them in Middle English and share them with others as they were meant to be heard. The only poem I have memorized at the moment (besides my own) is the opening to Chaucer's Canterbury Tales:

  Whan that Aprill with his shoures soote  

  The droghte of March hath perced to the roote

  And bathed every veyne in swich licour

  Of which vertu engendred is the flour;

  Whan that Zephyrus eek with his sweete breethe

  Inspired hath in every holt and heethe

  The tendre croppes, and the yonge sonne

  Hath in the Ram his half cours yronne,

  And smale fowles maken melodye,

  That slepen al the nyght with open ye

  (So priketh hem Nature in hir corages)

  Than longen folk to goon on pilgrimages 

I totally butcher the pronunciation, but I do my best and it feels so awesome on the tongue. I guess I prefer to memorize poetry by ear and mouth than by heart.