It appears that the simplest approximate algorithm one can implement (but with no minimality guarantees) is the one of this paper:
A fast and effective heuristic for the feedback arc set problem, by P. Eades, X. Lin, W.F. Smyth.
It is very easy to implement and works quite fast for large graphs (I tried it on a graph of 2.5 million edges and around 100 thousand nodes and broke all cycles in less than a minute).