In this work we investigate the wave-propagation properties of pure shear, inhomogeneous, viscoelastic plane waves in the symmetry plane of a monoclinic medium. In terms of seismic propagation, the problem is to describe SH-waves traveling through a fractured transversely isotropic formation where we assume that the waves are inhomogeneous with amplitudes varying across surfaces of constant phase. This assumption is widely supported by theoretical and experimental evidence.The results are presented in terms of polar diagrams of the quality factor, attenuation, slowness, and energy velocity curves. Inhomogeneous waves are more anisotropic and dissipative than homogeneous viscoelastic plane waves, for which the wavenumber and attenuation directions coincide. Moreover, the theory predicts, beyond a given degree of inhomogeneity, the existence of 'stop bands' where there is no wave propagation. This phenomenon does not occur in dissipative isotropic and elastic anisotropic media. The combination of anelasticity and anisotropy activates these bands. They exist even in very weakly anisotropic and quasi-elastic materials; only a finite value of Q is required. Weaker anisotropy does not affect the width of the bands, but increases the threshold of inhomogeneity above which they appear; moreover, near the threshold, lower attenuation implies narrower bands. A numerical simulation suggests that, in the absence of material interfaces or heterogeneities, the wavefield is mainly composed of homogeneous waves.