The diffusion of H-atoms is relevant for innumerous physical–chemical processes in metals. A detailed understanding of diffusion in a polycrystalline material requires the knowledge of the activation energies (ΔE_a’s) for diffusion at different defects. Here, we report a study of the diffusion of H-atoms at the Σ9 and Σ5 grain boundaries (GBs) of fcc Cu that are relevant for practical applications of the material. The complete set of possible diffusion pathways was determined for each GB and we compared the ΔE_a at bulk fcc Cu with the landscape of ΔE_a’s at these defects. We found that while a number of diffusion pathways at the GBs have high tortuosity, there are also many paths with very low tortuosity because of specific structural features of the interstitial GB sites. These data show that the diffusion of H-atoms at these GBs is highly directional but can be fast because at certain paths the ΔE_a can be as low as 0.05 eV. The lowest energy paths for diffusion of H-atoms through the whole GB models are ΔE_a = 0.05 eV for the Σ9 and ΔE_a = 0.20 eV at Σ5 which compare with ΔE_a = 0.42 eV for the bulk fcc crystal. This shows that H-atoms will be able to diffuse very fast at these defects. With the Laguerre–Voronoi tessellation method, we studied how the local atomic structure of the interstitial sites of the GBs leads to different ΔE_a’s for diffusion of H-atoms. We found that the volume expansions and the coordination numbers alone cannot account for the magnitude of the ΔE_a’s. Hence, we developed a symmetry quantifying parameter that measures the deviation of symmetry of the GB sites from that of the bulk octahedral site and hence accounts for the distortion at the GB site. Only when this parameter is introduced together with the volume expansions and the coordination numbers, it is possible to correlate the local structure with the ΔE_a’s and to obtain descriptors of diffusion. The complete set of data shows that the extrapolation of diffusion data for H-atoms between different types of GBs is non-trivial and should be done with care.