That's not too big.
so if you know which of the relationships in the cycle you want to remove it gets easier.
It would also be easier if you know the max-length of your potential cycles.
Is your graph fully connected? Or are there isolated subgraphs?
with cypher in 1.8 you can do:
start n=node(*) match n-[*0..]->x-[r:TYPE]->n delete r
with a traversal it is similar, you start at a node and traverse from there up until your max length or until you hit the start-node again (or another node that is already on your path) and your remove the last relationship to it.
Michael