Diffusion across surfaces generally involves motion on a vibrating but otherwise stationary substrate. Here, using molecular dynamics, we show that a layered material such as graphene opens up a new mechanism for surface diffusion whereby adsorbates are carried by propagating ripples in a motion similar to surfing. For water nanodroplets, we demonstrate that the mechanism leads to exceedingly fast diffusion that is 2-3 orders of magnitude faster than the self-diffusion of water molecules in liquid water. We also reveal the underlying principles that regulate this new mechanism for diffusion and show how it also applies to adsorbates other than water, thus opening up the prospect of achieving fast and controllable motion of adsorbates across material surfaces more generally.