Lots of languages, depending on exactly what you mean or are looking for. We have examples like your Somali examples in English: "middle of the road, outside of the town". Niger-Congo languages generally don't have prepositions and they widely use "N of N" constructions.
Technically, "preposition" is about half the story and "postposition" being the other half, the difference being whether the functional marker precedes or follows the nominal expression – I assume you don't really care about the order. Based on your examples, I surmise that you mean a particular subset of such functional markers, those that convey some kind of semantic relationship between the nominal clause and the sentence. Typically the first examples that come to mind express spatial relations, but also temporal, causal and possession relations.
In Bantu, as a well-studied example of Niger-Congo, the construction is generally expressed as "(modifier-N) link+ [Modified NP]", for example Shona pamusoro pe mvura "because of the rain", where mvura is "rain". This can be etymologically decomposed into an English clause as "on the head of the rain". The linker pe is basically an indicator that the second nominal (mvura) is syntactically subordinate to the functional head pamusoro, and it agrees in noun class with the clause-initial functional head. This is similar to but not the same as Somali, the difference being what triggers agreement (in Somali, agreement is with the lexical noun, not the functional head).
Another common construction used to express the relations that are conveyed in English by prepositions is the serial verb construction. In this construction, (apparently) single clauses contain two (or more) verbs, one of which conveys the core propositional meaning and the other of which encodes "other information" including what would be prepositions in English. Example: Yoruba mo fi ade ge igi lit. "I took machete cut wood", meaning "I cut the wood with a machete" – here, "take" is used for English "with" in the instrumental sense.