This manuscript presents a new nonlocal homogenization model (NHM) for wave dispersion and attenuation in elastic and viscoelastic periodic layered media. Homogenization with multiple spatial scales based on asymptotic expansions of up to eighth order is employed to formulate the proposed nonlocal homogenization model. A momentum balance equation, nonlocal in both space and time, is formulated consistent with the gradient elasticity theory. A key contribution in this regard is that all model coefficients including high-order length-scale parameters are derived directly from microstructural material properties and geometry. The capability of the proposed model in capturing the characteristics of wave propagation in heterogeneous media is demonstrated in multiphase elastic and viscoelastic materials. The nonlocal homogenization model is shown to accurately predict wave dispersion and attenuation within the acoustic regime for both elastic and viscoelastic layered composites.