"Jazz chants," or rhythmic singing, are a time-tested method recommended by ESL teachers to help students learn and remember new material.

For example, I just made up these simple rhyming chants to help people learn the singular endings for "type 1 feminine" nouns (those that have -а/-я in the nominative singular), and also to help you remember the typical functions of each case:

"Nom-nom-nom!" said the девушка, as she ate up all the baklava!
Gay men generally flee from a девушки!
Commander Data gave the tray to the девушке!
I accuse you, wicked девушку!
The ol' cowboy with his девушкой played an instrumental version of "Ode to Joy!"
I heard a rumor today about the preppy девушке!

(Nominative: subject of "said" and "ate up")
(Genitive: shows movement away from)
(Dative: indirect object, giving X to Y)
(Accusative: direct object of the verb "to accuse")
(Instrumental: shows "with" or togetherness)
(Prepositional: when speaking "about" something.)

Of course, the genitive case has other functions besides "separation or movement away"! However, the point of this chant is to trigger your memory with an example of how to use the genitive, but not to explain every possible use of the genitive.

And needless to say, it doesn't have to be great poetry (or great jazz). Just find some rhymes that help you associate the sound of the case endings (such as -ой, fem. sing. instrumental) with the meaning/function of the cases ("the boy was with her").