In response to a question by James Breeze, I mentioned that the case of the noun or pronoun following a preposition depends on which preposition it is. Since then, I found the following list:

Prepositions followed by noun or pronoun in accusative case:
pe on; la at, to; lângă near; sub under; cu with; fără without; după after; spre towards; în in, din from; până until; despre about; către to; prin through; de of since, for

Prepositions followed by noun or pronoun in dative-genitive case:
datorită due to; mulţumită thanks to; graţie thanks to; conform according to; contrar contrary to; asemenea similarly to, aspura on, about; contra against; deasupra above; dedesubtul under, împotriva against; înaintea before.

Notes:
1. The source is Ramona Gönczöl-Davies, Romanian: An Essential Grammar, London and New York, Routledge, 2008. I downloaded the pdf from the web. FYI this seems to me to be to be a good book, with a clear presentation of the basics of Romanian grammar. I invite suggestions for other grammars.
2. The above book uses the five case system, which presents as two distinct cases dative and genitive. Since these two cases have the same form, I think that they should be regarded as a single case, and so I have presented them as such. 
3. In addition to the above simple propositions, the book also presents compound prepositions, and prepositional phrases (which seem to me to be compound prepositions).

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