The acoustic response of a source in an anisotropic viscoelastic medium has not only a directionally dependent speed of propagation, but also a directionally dependent rate of attenuation. The long wavelength properties of a layered composite comprised of two alternating constituent layers, both of which are isotropic, but with one elastic and the other viscoelastic (modeled as a ``standard solid''), are investigated. Such a structure behaves as a homogeneous transversely isotropic viscoelastic medium. The plane-wave phase slowness and quality factor as a function of direction are shown for three cases. The first two are (a) when the slower medium is lossy and (b) when the faster medium is lossy. In these cases, the curves for loss versus angle are highly directional, but the actual patterns depend dramatically upon the choice of (a) or (b). The third case examines the behavior when the velocities of the constituents are approximately the same, but the loss is such that which constituent is fast and which is slow is dependent on frequency.