With a view toward explaining measurements of ultrasonic attenuation in metals, the coefficient of shear viscosity of a free electron gas interacting with thermal phonons and local crystal inhomogeneities is computed. The methods employed are essentially those developed for the problem of electrical conductivity. By means of a variational principle, a general formal solution is obtained. The effective mean free path is found to be somewhat smaller and of a somewhat more complicated (though not qualitatively different) temperature dependence than that associated with electrical conduction. The effect of a transverse magnetic field is determined for the case when a time of relaxation exists.