  Let $G = (V, E)$ be an $n$-vertex edge-colored graph. In 2013, H. Li proved
that if every vertex $v \in V$ is incident to at least $(n+1)/2$ distinctly
colored edges, then $G$ admits a rainbow triangle. We prove that the same
hypothesis ensures a rainbow $\ell$-cycle $C_{\ell}$ whenever $n \ge 432 \ell$.
This result is sharp for all odd integers $\ell \geq 3$, and extends earlier
work of the authors for when $\ell$ is even.
