  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 establish a corresponding
result for fixed even rainbow $\ell$-cycles $C_{\ell}$: if every vertex $v \in
V$ is incident to at least $(n+5)/3$ distinctly colored edges, where $n \geq
n_0(\ell)$ is sufficiently large, then $G$ admits an even rainbow $\ell$-cycle
$C_{\ell}$. This result is best possible whenever $\ell \not\equiv 0$ (mod 3).
Correspondingly, we also show that for a fixed (even or odd) integer $\ell \geq
4$, every large $n$-vertex oriented graph $\vec{G} = (V, \vec{E})$ with minimum
outdegree at least $(n+1)/3$ admits a (consistently) directed $\ell$-cycle
$\vec{C}_{\ell}$. Our latter result relates to one of Kelly, K\"uhn, and
Osthus, who proved a similar statement for oriented graphs with large
semi-degree. Our proofs are based on the stability method.
