Another thing: The "jazz chants" can be much simpler than the examples I gave above.

К примеру, suppose you want to memorize the endings for singular masculine nouns.

Whether you love Lady Gaga, hate her, or really really hate her, I assume that most people have seen the video for "Bad Romance", in which she marches around in "armadillo shoes" (aka "lobster-claw shoes") and chants:

Walk, walk, fashion, baby,
Work it, move that bitch ca-razy!
Notice that "walk, walk, fa-shion, ba-by" has шесть слогов (six syllables). And, conveniently, there are six noun cases in modern Russian: nominative, genitive, dative, accusative, instrumental, prepositional.

So, we can take a Russian noun like волк (wolf) -- an example that I chose because it's masculine and animate, and also happens to sound rather similar to the English word "walk." The singular declension is:

волк (Nom)
волка (Gen)
волку (Dat)
волка (Acc)
волком (Inst)
волке (Prep)

And then we can imagine Lady Gaga in her armadillo shoes, with a wolf on a leash, singing:

"Walk, walk, fa-shion ba-by
Work it, move that bitch, cah-razy!
Волк! -а! -у! -а! -ОМ! -е!
Russian grammar makes me come, hey!
Волк! -а! -у! -а! -ОМ! -е!
Russian grammar makes me come, hey!
Волк! -а! -у! -а! -ОМ! -е!
I'm a free bitch, baby!"

Or, alternatively, you can imagine the волк catching Little Red Gaga Hood ("Красная Гагочка"?) and gobbling her up (because she can't run in those ridiculous shoes), and then she sings this from inside the wolf's tummy...

This example doesn't help you to remember how the different cases function in sentences -- but if you're already familiar with the difference between "dative" and "accusative," and assuming you've learned them in the order N-G-D-A-I-P, then the Lady Gaga example can help you remember the case endings in order for a noun like волк.