In ancient Greek prepositions could govern the genitive, the dative or the accusative. Often the same preposition could govern more than one cases leading to different meanings.
In modern Greek things are much simplified and almost all prepositions govern the accusative. However there are still some prepositions and mostly adverbs that govern the genitive. Some of them are a bit archaic and they are used in high language only. Here is a list of them:
a. Modern Greek prepositions
The only modern Greek preposition (i.e. not sounding archaic) that governs the genitive is κατά. Actually it governs both the genitive and the accusative giving different meanings:
κατά + gen.: against
είμαι κατά του φεμινισμού - I'm against feminism
μέτρα κατά των ναρκωτικών - measures against the drugs
κατά + acc.: at around, according to
κατά τις πέντε - at around five o' clock
κατά τον Αριστοτέλη - according to Aristoteles
b. Adverbs
λόγω: because (of)
δεν ταξιδέψαμε λόγω κακοκαιρίας - we didn't travel because of the bad weather
λόγω του καύσωνα - because of the heat wave
λόγω του ότι δεν ήρθε - because he didn't come
εξαιτίας: because of
εξαιτίας του Γιώργου - because of George
εξαιτίας σου - because of you
εναντίον: against
ο πόλεμος εναντίον της Ινδίας - the war against India
η Άρσεναλ εναντίον της Μάντσεστερ - Arsenal against Manchester
μεταξύ: between, among
μεταξύ των ανθρώπων - among people
μεταξύ Αθήνας και Ρώμης - between Athens and Rome
μεταξύ μας - among us
μέσω:through
πήγαμε στη Σόφια μέσω Θεσσαλονίκης - we went to Sofia through Salonica
c. High language prepositions
περί: about
μιλούσαν περί πολιτικής - they were talking about politics
οι απόψεις του περί ηθικής - his opinions about ethics
επί: on, about
επί του θέματος αυτού - on that issue
δια: through
δια της φορολόγησης θα συγκεντρωθούν τα χρήματα - through taxes money will be gathered