These prepositions always take accusative case (den, das, die, die):

durch

 für

 gegen

 ohne

 um

They are easy to remember because they can be sung to the tune of raindrops keep falling on my head.

These prepositions always take dative case (dem, dem, der, den).

aus

außer

bei

mit

nach

seit

von

zu

They are easy to remember because they can be sung to the tune of blue danube.

These prepositions take accusative or dative case.  

If you remember the prepositions above, you don't have to remember these (if they are not on one of the previous lists, they are probably on this list).  However, these prepositions can be sung to the tune of ode to joy.

an

auf

hinter

in

neben

über

unter

vor

zwischen

The prepositions govern accusative case if the prepositional phrase modifies a verb of motion:

            Die Katze läuft unter den Tisch.

The prepositions govern dative case if the prepositonal phrase modifies a stationary verb:

            Die Katze schläft unter dem Tisch.

When the motion vs. stationary is irrelevant, the prepositions usually take accusative (e.g. denken an + acc; sich freuen auf)

Finally, there is a small group of prepositions tht takes the genitive case (des, des, der, der)

(in many German dialects, in some dialects they take the dative case, but that won't work as an excuse on a test!)

(an)statt  (instead of)

trotz  (in spite of)

während  (during)