The Cell Method is a recently developed numerical method that is giving interesting results in several fields of physics and engineering. In this paper, first a brief description of the method for elasticity problems is given and successively the elastodynamics formulation is derived. The method leads to an explicit solution system, combining the advantages of a diagonal mass matrix and the possibility of using unstructured meshes. The convergence rate has been tested in reference to the problem of free harmonic vibrations in a system with one degree of freedom, showing that the Cell Method has the same convergence rate of II order Runge Kutta method, but its accuracy is better. The Cell Method results in 2D and 3D have been compared with those obtained with the commercial codes ANSYS and ABAQUS in the problem of the longitudinal vibration of a bar with free ends, for which the exact analytic solution is found in literature. The Cell Method results are comparable with or better than those obtained with FEM, and they are particularly interesting from the point of view of computation time and memory requirements for very large meshes.