The orbits of stars close to a massive black hole (MBH) are nearly Keplerian ellipses. Such orbits exert long-term torques on each other, which lead to an enhanced angular momentum relaxation known as resonant relaxation. Under certain conditions, this process can modify the angular momentum distribution and affect the interaction rates of the stars with the MBH more efficiently than non-resonant relaxation. The torque on an orbit exerted by the cluster depends on the eccentricity of the orbit. In this paper, we calculate this dependence and determine the resonant relaxation time-scale as a function of eccentricity. In particular, we show that the component of the torque that changes the magnitude of the angular momentum is linearly proportional to eccentricity, so resonant relaxation is much more efficient on eccentric orbits than on circular orbits.