Analytic energy gradients with respect to nuclear coordinates for an exact exchange-only (EXX) Kohn-Sham method are presented. In the underlying EXX method, the exact exchange potential is obtained as the electrostatic potential of an exchange charge density, which is determined via the optimized effective potential method. Parts of the presented calculation of analytic EXX energy gradients can be reused for analytic energy gradients in self-consistent Kohn-Sham methods treating correlation via the adiabatic-connection fluctuation-dissipation theorem, e.g., methods relying on the random phase approximation. A version of the analytic EXX energy gradients that uses density-fitting is shown to be highly efficient. The accuracy of the analytic energy gradients is tested by comparison with numerically calculated gradients.