New measurements of the attenuation of acoustic waves in quartz have been obtained from surface excitation in a novel type of resonant cavity configuration. These are believed to be more accurate than previously published values. The attenuation is found to be proportional to the square of the frequency over the range 300 to 3000 Mc/s at a temperature of 30 $^\circ$C. A general theory, involving a viscosity tensor, is developed for the calculation of the attenuation of acoustic waves in crystalline solids. Symmetry of the viscosity matrix is determined by the crystal class and is the same as that of the elastic constant matrix. Numerical values for the viscosity coefficients are specific to the material and from a knowledge of these the attenuation of a given mode in a selected direction of propagation can be calculated. The validity of this analysis is confirmed by the results of the attenuation measurements in quartz and is further supported by data for silicon.