The paper presents a spectral analysis of wave propagation in bounded two-dimensional elastic media discretized under assumptions of two selected local computational strategies. The Local Interaction Simulation Approach and Cellular Automata for Elastodynamics are first overviewed with particular emphasis on their fundamental assumptions, treatment of boundary conditions, and resulting formulations. Each method’s numerical dispersion characteristics for bulk waves are then analyzed using a closed set of iteration equations, after which guided Lamb wave dispersion characteristics are analyzed for homogeneous and layered plates. In all cases, comparisons to finite element dispersion characteristics, re-formulated in the form of local iteration equations, provide insight into the relative advantages and disadvantages of the local formulations and reveal non-physical effects induced by numerical discretization. Finally, verification of the analytical dispersion results is presented using transient simulations with accompanying spectral analysis.