An experimental technique has been developed to determine the surface energy anisotropy of crystalline solids. The technique is based on atomic force microscopy measurements, which are used to quantify the geometry of thermal grooves, and electron backscattered diffraction pattern measurements, which are used to specify crystallographic orientations. Observations are made at circumferential thermal grooves, where it is assumed that Herring's local equilibrium condition for a triple junction holds and that the grain-boundary energy is independent of its boundary plane. A truncated double Fourier series is used to approximate the surface energy, and the unknown coefficients of the series are determined by fitting the observations to the local equilibrium condition. The method, which should be applicable to most polycrystalline materials, has been tested on magnesia that has been thermally grooved at 1400 degrees C in air. The maximum of the best-fit surface energy function is at (111) and the minimum is at (100), The relative surface energies of the low-index planes are gamma(110)/gamma(100) = 1.040 +/- 0.008 and gamma(111)/gamma(100) = 1.072 +/- 0.010.