Biot’s equations of poroelasticity were solved to study the effects of fracture connectivity on S-wave attenuation caused by wave-induced fluid flow at the mesoscopic scale. The methodology was based on numerical quasistatic pure-shear experiments performed on models of water-saturated rocks containing pairs of either connected or unconnected fractures of variable inclination. Each model corresponded to a representative elementary volume of a periodic medium. Inertial terms were neglected, and hence, the observed attenuation was entirely due to wave-induced fluid flow at the mesoscopic scale. We found that when fractures are not connected, fluid flow in the embedding matrix governs S-wave attenuation, whereas fluid flow through highly permeable fractures, from one fracture into the other one, may dominate when fractures are connected. Each of these energy-dissipation phenomena has a distinct characteristic frequency, with the S-wave attenuation peak associated with flow through connected fractures occurring at higher frequencies than that associated with flow in the embedding matrix. Exploring a range of geometric arrangements of either connected or unconnected fractures at different inclinations, we also observed that the magnitude of S-wave attenuation at both characteristic frequencies shows a strong dependence on fracture inclination. For comparison, we performed quasistatic uniaxial compressibility tests to compute P-wave attenuation in the same models. We found that the attenuation patterns of S-waves tend to differ fundamentally from those of P-waves with respect to fracture inclination. The attenuation characteristics of P- and S-waves in fractured media are thus, largely complementary. With respect to fracture connectivity, we observed that S-wave attenuation tends to follow a specific pattern, indeed, more consistently than that of the P-waves. Our results point to the promising perspective of combining estimates of attenuation of P- and S-waves to infer information on fracture connectivity as well as on the effective permeability of fractured media.