We study the evolution of grain boundary networks by the mean-curvature flow under the restriction that the networks are Voronoi diagrams for a set of points. For such evolution we prove a rigorous universal upper bound on the coarsening rate. The rate agrees with the rate predicted for the evolution by mean-curvature flow of the general grain boundary networks, namely that the typical grain area grows linearly in time. We perform a numerical simulation which provides evidence that the dynamics achieves the rate of coarsening that agrees with the upper bound in terms of scaling.