The linear theory of viscoelasticity is used to study the effects of anelasticity on seismic SH body waves propagating through a layered homogeneous isotropic medium. We present a solution of the wave propagation problem in terms of asymptotic ray theory. The geometrical spreading factor for an arbitrary ray generated by a point source close to the surface of a plane-layered anelastic medium is calculated and is found to vary with the direction of maximum attenuation of the initial ray segment emanating from the source. Combining anelastic reflection and transmission coefficients and the geometrical spreading factor, ray-synthetic seismograms for SH body waves are computed for a plane-layered crustal model in both the elastic and anelastic cases. The seismograms for the anelastic cases exhibit amplitude attenuation and waveform spreading. The amplitudes and phases of the arrivals are also seen to depend significantly on the direction of maximum attenuation of the initial ray segment. This is primarily due to the nature of the reflection and transmission coefficients.