Honeycomb structure with phase change material (PCM) is frequently used in passive thermal management devices. The geometrical shape of the honeycomb core greatly influences the melting rate of the PCM. This paper investigates the melting rates of PCM in honeycomb cores of non-hexagonal cells in comparison with that of hexagonal cell in three Rayleigh numbers. The objective is to find the optimal shape in order to reduce the melting time of the PCM. The constrained melting behaviors of PCM in triangular, quadrilateral, hexagonal, and circular honeycomb cores are numerically studied. The enthalpy porosity technique and finite volume method are used in this paper. The instantaneous liquid fraction and energy absorption of PCM in different honeycomb cores are discussed in detail. The influences of the placed orientation and aspect ratio of different cores on melting rates of PCM are considered. Results show that the melting rate of PCM in a rectangular core is always higher than the hexagonal core for the given aspect ratio and Rayleigh number. The geometrical factor (GF), which indicates the cross-sectional area per unit perimeter, is found to be an important index on the melting rate. At a small Rayleigh number, it takes a longer melting time of the PCM for the core with a larger GF. As the Rayleigh number is large, the melting time of the PCM is affected by both the GF and the orientation of the cores.