Wave energy can be dissipated gradually when it is propagated in viscoelastic damping material (VDM) composite structures. In this paper, after the specimens with different opening ratios (ORs) of VDM layer are prepared, the elastic wave energy propagation and dissipation characteristic of periodically perforated VDM cored sandwich structures are investigated by an experimental method. The sandwich structures are discretized into several testing points in our experiment. When the complex velocity, equivalent effective mass, and external excitation forces have been obtained at each testing point by sensors, the energy dissipation in the sandwich structure is determined based on the energy dissipation mechanism of wave transmission in solid. The experimental results are then compared with theoretical and numerical simulation results. By analyzing the computational accuracy of theoretical and numerical results using experimental data, it is shown that high consistency between theoretical, numerical, and experimental results can be achieved, especially in the medium-frequency and high-frequency ranges. Thus, our experimental results demonstrate that periodically perforated VDM sandwich structures can be applied to engineering practice for their good performance of dissipation characteristics in the middle-frequency and high-frequency ranges.