Sandwich panels are widely used in many engineering applications where saving weight while maintaining high strength and stiffness is required. The most common core structure in sandwich panels is the two-dimensional regular hexagonal cell shape, denoted as Honeycomb. In recent times, bioinspired materials and structures have become increasingly attractive to researchers, as they provide adequate functional properties. The goal of the present work is to study two new bioinspired structures aimed at improving the performance of sandwich panel cores. Among all the large amount of structures that nature provides, two novel cores inspired in the structures of enamel and of bamboo were chosen. The compressive and flexural properties of these two innovative cellular structures were assessed and compared with the classic honeycomb. All the arrangements were numerically simulated for different relative densities. The fused deposition modelling technique enables to print selected samples in polylactic acid that were experimentally tested in compression and in bending. Results show that the mechanical properties depend strongly on the core geometry, on the relative density and on the cell thickness distribution. A satisfactory agreement was found between finite element results and experimental data. For the same relative density, the bioinspired natural structures proposed in the present study are potential competitors with the traditional core structures in what concerns strength, stiffness and energy absorption.