The propagation of elastic waves in a periodic laminate is considered. The stratified medium is modeled as a homogenized material where the stress depends on the strain and additional higher order strain gradient terms. The homogenization scheme is based on a lattice model approximation tuned on the dispersive properties of the real laminate. The long-wave asymptotic approximation of the model shows that, despite the simplicity of the parameters identification, the proposed approach agrees well with the exact solution in a wide range of elastic impedance contrasts, also in comparison with different approximations. The effect of increasing order of approximation is also investigated. A final example of a finite structure under an impact excitation proves that the model behaves well when applied in the transient regime and that it can be considered a simple but consistent approach to build efficient algorithms for the numerical analysis of elastodynamics problems.