A Finite Volume Meshless Local Petrov-Galerkin (FVMLPG) method is presented for elastodynamic problems. It is derived from the local weak form of the equilibrium equations by using the Finite Volume (FV) and the Meshless Local Petrov-Galerkin (MLPG) concepts. By incorporating the moving least squares (MLS) approximations for trial functions, the local weak form is discretized, and is integrated over the local subdomain. Numerical examples for solving the transient response of the elastic structures are included.