Based on the moving least-squares (MLS) approach, an efficient meshless method is employed to generate the displacement functions for vibration analysis of elastic bodies. The equation of motion is established by following the standard procedure and the boundary conditions are imposed by applying penalty functions. As the displacement functions are expressed in terms of weight functions, the accuracy will depend on the parameters of the weight functions. Therefore, a parametric study is carried out to determine the best values for these parameters. To demonstrate the accuracy, modal analyses of the beams and plates with different boundaries have been carried out. In addition, the responses of these structures under dynamic excitation have been analyzed. The examples include simply supported beams subjected to sudden excitations and simply supported plates subjected to initial displacements.