We show that on a random infinite graph G of polynomial growth where simple random walk is stationary, it is diffusive along a subsequence, i.e., the second moment of the distance from the starting point grows at most linearly in time. This extends a result of Kesten that applied to the extrinsic metric on subgraphs of the lattice Zd, and answers a question due to Benjamini, Duminil-Copin, Kozma...