The buckling and postbuckling responses of viscoelastic-layered composites are investigated using finite-element simulations. These composites consist of alternating layers of a stiff elastic constituent and of a soft viscoelastic constituent. In response to compressive loads in the layer direction, elastic instabilities significantly affect the finite deformation mechanics of these composites. The dependence of the critical strain and critical wavenumber on strain rate is analyzed. In the postbuckling regime, the wavenumber of the mode of deformation is found to be highly dependent on strain rate and time and can be used to identify three different regimes that depend on the volume fraction of the stiff constituent. Interestingly, a transition from a wrinkling mode to a longwave mode can be observed when the strain rate is varied for moderate volume fractions of the stiff material. Analytical formulae for the buckling and postbuckling of the elastic-layered composites are used to interpret numerical results obtained for viscoelastic-layered composites. Viscoelastic-layered composites exhibit a wide range of rate-dependent mechanical behavior and could have applications in vibration damping and acoustic metamaterials.