Honeycomb structures, owing to their microstructural periodicity, exhibit unique and complex acoustic properties. Tuning their acoustic properties typically involves either changing their topology or porosity. The former route can lead to topologies that may not be readily amenable for large-scale production, while the latter could negatively affect the honeycombs’ weight. An ideal approach for tailoring the acoustic behavior of honeycombs should neither affect their porosity nor should they require customized and expensive fabrication methods. In this work, a novel honeycomb design that alters the microstructural topological features in a relatively simple way, while preserving the porosity of the honeycombs, to tune the acoustic properties of the honeycombs is proposed. The proposed honeycomb can be fabricated using the traditional approach employed to mass produce honeycomb structures; that is by bonding identical corrugated sheets with two periodic thicknesses. The acoustic behavior of the proposed honeycomb in terms of dispersion and phase velocities is analyzed using the finite element method. Simulation results demonstrate the potential of the designed honeycomb to exhibit tailored acoustic behavior at a constant porosity or mass. For example, it is demonstrated that the phase velocities of asymmetric and symmetric waves traversing the proposed honeycomb of aluminum with 90% porosity can be tuned by 30% and 17%, respectively.