  An oriented cycle is an orientation of a undirected cycle. We first show that
for any oriented cycle $C$, there are digraphs containing no subdivision of $C$
(as a subdigraph) and arbitrarily large chromatic number. In contrast, we show
that for any $C$ a cycle with two blocks, every strongly connected digraph with
sufficiently large chromatic number contains a subdivision of $C$. We prove a
similar result for the antidirected cycle on four vertices (in which two
vertices have out-degree $2$ and two vertices have in-degree $2$).
