This won't help you with all the prepositions, particularly as some of them can take several cases, but as a general rule:
- the genitive is used to express separation or motion away from a point
- the dative is used to express location
- the accusative is used to express motion towards a point
The only other suggestion I can make is to find a good, clear example of the use of each preposition and memorize the entire phrase. If you can recognize the case endings on the noun, you know what case the preposition is being used in (this probably works a little better for languages you actually speak -- I still use the technique when I can't remember a gender in German -- but in theory it ought to work for Greek also).
IPHIGENIE: Kann uns zum Vaterland die Fremde werden?
ARKAS: Und dir ist fremd das Vaterland geworden.
IPHIGENIE: Das ist's, warum mein blutend Herz nicht heilt.
(Goethe, Iphigenie auf Tauris)