Water around biomolecules slows down with respect to pure water, and both rotation and translation exhibit anomalous time dependence in the hydration shell. The origin of such behavior remains elusive. We use molecular dynamics simulations of water dynamics around several designed protein models to establish the connection between the appearance of the anomalous dynamics and water-protein interactions. For the first time we quantify the separate effect of protein topological and energetic disorder on the hydration water dynamics. When a static protein structure is simulated, we show that both types of disorder contribute to slow down water diffusion, and that allowing for protein motion, increasing the spatial dimentionality of the interface, reduces the anomalous character of hydration water. The rotation of water is, instead, altered by the energetic disorder only; indeed, when electrostatic interactions between the protein and water are switched off, water reorients even faster than in the bulk. The dynamics of water is also related to the collective structurea voir the hydrogen bond (H-bond) networkformed by the solvent enclosing the protein surface. We show that, as expected for a full hydrated protein, when the protein surface offers pinning sites (charged or polar sites), the superficial water-water H-bond network percolates throughout the whole surface, hindering the water diffusion, whereas it does not when the protein surface lacks electrostatic interactions with water and the water diffusion is enhanced.