  We prove that every family of (not necessarily distinct) odd cycles $O_1,
\dots, O_{2\lceil n/2 \rceil-1}$ in the complete graph $K_n$ on $n$ vertices
has a rainbow odd cycle (that is, a set of edges from distinct $O_i$'s, forming
an odd cycle). As part of the proof, we characterize those families of $n$ odd
cycles in $K_{n+1}$ that do not have any rainbow odd cycle. We also
characterize those families of $n$ cycles in $K_{n+1}$, as well as those of $n$
edge-disjoint nonempty subgraphs of $K_{n+1}$, without any rainbow cycle.
