The reconfiguration graph $R_k(G)$ for the $k$-colorings of a graph $G$ has as vertices all possible $k$-colorings of $G$ and two colorings are adjacent if they differ in the color of exactly one vertex. We use a list coloring technique inspired by results of Thomassen to prove that for a planar graph $G$ with $n$ vertices, $R_{10}(G)$ has diameter at most $8n$, and if $G$ is triangle-free, then $R_7(G)$ has diameter at most $7n$.