Many biological materials have outstanding properties unparalleled by artificial materials because of their periodic and/or hierarchical structures. In this paper, the longitudinal elastic wave propagation in nacre-like materials with brick-and-mortar microstructures has been studied theoretically by taking the different deformation mechanisms into account. A viscoelastic model in relation to the sacrificial bond behavior for the organic matrix has been developed. The shear-lag model is adopted to simulate the deformation of the brick-mortar structure. Complex band structures have been analyzed and the mechanical model has been validated by finite element method. Results have shown that bandgaps with lower edge close to vanishing frequency may arise in such bio-inspired periodic structures and attenuated propagation of elastic waves may be dominant in the range of higher frequencies. It may cast light on the dynamic strengthening mechanisms in nacre-like materials and provide a scientific basis for the design of bio-inspired metamaterials with low-frequency bandgaps.