Bandgaps in the dispersion curves of acoustic metamaterials are of great scientific and engineering importance for developing novel materials in vibration-isolation and noise-reduction applications. In this work, two dimensional achiral and chiral composite metamaterials are studied for their bandgap properties in relation to microstructures and physical properties of constituents. In particular, parametric studies have been conducted to investigate the effects of the Young’s modulus and density mismatches, as well as thickness of ligaments on bandgaps. When the ratio of matrix density to skeleton density ρm/ρs less than 1, largest bandgaps can be found in the modulus mismatch Em/Es = 1 chiral case, not in the Em/Es = 100 cases, indicating the interplay among chirality, modulus and density on bandgap structures in the metamaterials. When chirality, modulus and density are given, the thickness of ligaments provides additional degrees of freedom to design bandgaps.