In the current work, we elaborate upon a beam mechanics-based discrete dynamics approach for the computation of the dispersion characteristics of periodic structures. Within that scope, we compute the higher order asymptotic expansion of the forces and moments developed within beam structural elements upon dynamic loads. Thereafter, we employ the obtained results to compute the dispersion characteristics of one- and two-dimensional periodic media. In the one-dimensional space, we demonstrate that single unit-cell equilibrium can provide the fundamental low-frequency band diagram structure, which can be approximated by non-dispersive Cauchy media formulations. However, we show that the discrete dynamics method can access the higher frequency modes by considering multiple unit-cell systems for the dynamic equilibrium, frequency ranges that cannot be accessed by simplified formulations. We extend the analysis into two-dimensional space computing with the dispersion attributes of square lattice structures. Thereupon, we demonstrate that the discrete dynamics dispersion results compare well with that obtained using Bloch theorem computations. We show that a high-order expansion of the inner element forces and moments of the structures is required for the higher wave propagation modes to be accurately represented, in contrast to the shear and the longitudinal mode, which can be captured using a lower, fourth-order expansion of its inner dynamic forces and moments. The provided results can serve as a reference analysis for the computation of the dispersion characteristics of periodic structural systems with the use of discrete element dynamics.