A Meshless Local Petrov-Galerkin (MLPG) method based on Local Boundary Integral Equation (LBIE) techniques is employed here for the solution of transient elastic problems with damping. The Radial Basis Functions (RBF) interpolation scheme is exploited for the meshless representation of displacements throughout the computational domain. On the intersections between the local domains and the global boundary, tractions are treated as independent variables via conventional boundary interpolation functions. The MLPG(LBIE)/RBF method is applied to both transient and steady-state Fourier transform elastodynamic domains. In both cases the LBIEs employ the simple elastostatic fundamental solution instead of the complicated time and frequency dependent ones. The transient version of the present MLPG(LBIE)/RBF technique utilizes the$\theta$-Wilson finite difference scheme for the treatment of acceleration and velocity terms, while the frequency domain formulation exploits the Fast Fourier Transform (FFT) for the conversion of frequency domain solutions into time domain fields. The accuracy of the proposed methodology is assesed with three representative numerical examples.