Sensitive mechanical or electrical components often require protection from the potentially damaging effects of vibration and shock loading. High-damping viscoelastic materials are usually used in the design of impact-absorbent components. Since shock transients are characterized by a broad frequency spectrum, it is imperative to properly model frequency dependence of material parameters over the frequency range of interest. The Anelastic Displacement Fields (ADF) method is used to incorporate frequency-dependence within a finite element formulation. This method considers the effect of material anelasticity on the displacement field, as opposed to directly modeling physical damping mechanisms. ADF-based, plane-stress, and plane-strain finite elements are developed in order to facilitate the modeling of complex viscoelastic structures. The governing equations and assumptions underlying the various finite element developments are presented. In this paper, corresponding finite element models are used to model shock propagation and absorption through viscoelastic beams. The model predictions are validated against wave propagation theory, which shows that ADF-based finite element models are capable of capturing wave propagation phenomena, such as geometric dispersion, and viscoelastic attenuation and dispersion of longitudinal waves in beams. The behavior of mechanical filters in realistic shock conditions is also investigated. ADF three-dimensional finite element models could thus be successfully employed to design mechanical filters, or compare the benefits of using one viscoelastic material over another for a given shock-mitigating task.