The anisotropy of friction on graphitic surfaces is investigated by a combined friction force microscopy and modeling study. Friction vectors deviate up to 15° from pulling directions. The strongest deviations are found for pulling directions which lie almost along one zigzag direction of the honeycomb structure, the preferred sliding direction on graphite surfaces and epitaxial graphene grown on SiC(0001). Atomic stick-slip events along and across molecular rows determine direction and magnitude of friction. Simulation and modeling reveal the role of temperature and of the two-dimensional character of the surface potential for the friction anisotropy.