This study presents the design of a locally resonant absorbing metamaterial in the form of cubes and rectangular cubes, formed from square and honeycomb shaped unit cells. In this context, we fabricated a box consisting of acoustic metamaterial panels to validate our numerical simulation model. Then, we developed simulation models of acoustic metamaterials with square and honeycomb unit cells, forming cubic and rectangular cube sound barriers. Our purpose was to verify the operating frequency range of the designed metamaterial and then, compare the behavior of the different acoustic metamaterial structures in the specified frequency range. The acoustic metamaterials proved to be significantly effective in sound wave absorption providing transmission losses of up to 17.5 dB, more or less similarly in the three different structures, due to the identical resonators used in all of the structures, despite difference in unit cell shapes, number of resonators in the unit cell, and geometry of the barrier structure (cube or rectangular cube). These perforated metamaterial structures were then, tested in a waveguide consisting of panels of locally resonating metamaterials. Simulation results indicated that significant stop band behavior appeared for both square and honeycomb structures in the frequency of 750–1200 Hz, with a transmission loss of up to 60 dB. A clear comparison with a regular plate sound insulator tuned to the same resonance frequency is made, and the bandwidth and transmission loss is compared under the constraints of same occupied space and same material type. According to the procedure described in this paper, it is possible to design stop bands with excellent performance in a certain targeted frequency range.