A \(k\)-edge coloring \(c\) of the edge set \(E (G)\) of a graph \(G\) is a surjective mapping \(c : E (G) \to [k] = \{1, 2, \ldots, k\}\). If \(\mathcal{F}\) and \(\mathcal{H}\) are families of graphs, \(MRS(K_n; \mathcal{F}, \mathcal{H})\) is the set of numbers \(k\) such that there is a \(k\)-edge coloring of \(K_n\) with respect to which there is neither a monochromatic copy of any \(F \in \mathcal{F}\) nor a rainbow copy of any \(H \in \mathcal{H}\) in \(K_n\). Our main result is that for all \(n \geq 2\), \(MRS(K_n;\{\text{odd cycles}\},\{\text{cycles}\}) = \{\lceil \log_2 n \rceil, \ldots, n – 1\}\). The proof will exploit an idea for edge-coloring connected graphs so as to forbid rainbow cycles to be found in [4].