Why does German have no memorizing tricks? If there are any please post them here.
The words: aus, bei, mit, nach, zeit, von, zu always take dative. The words: durch, für, gegen, ohne, um always take akkusativ. The words: an, auf, hinter, in, neben, über, unter, vor, zwischen take both akkusativ and dativ depending on the situation. Akkusativ when movement or an action is taking place, dative when location is the essence.
There are probably some mnemonics for these, but we simply learnt them by heart in school. Luckily when you have learnt the declination and a few rules such as these, your german understanding will be sufficient to do quite well.
I don't know of any songs or jingles. There are simple tricks, though. For example, any noun ending in -ung (e.g. Beziehung, Leistung, Umdrehung), -keit (e.g. Fähigkeit, Aufmerksamkeit), and -tion (e.g. Revolution, Kommunikation) will be feminine. Also a verb can become a noun and then has a neuter definite article. For example, the verb essen (to eat) as a noun is das Essen (the food or meal) and, the verb rennen (to run or race) as a noun is das Rennen (the race). I hope that this is of some help to you.