The negative Poisson's ratio (NPR) structures exhibit some unusual but highly useful mechanical properties. In comparison to two-dimensional (2D) NPR structures, less attention has been paid to the studies on three-dimensional structures due to the complex manufacturing process. In this paper, based on the two-dimensional nonconvex hexagonal cell, a novel three-dimensional (3D) NPR energy-absorbing structure is proposed. According to the beam theory, the equivalent Young's modulus and Poisson's ratio are deduced. Subsequently, numerous finite element (FE) simulations and compression tests were performed and a good agreement was observed among the theoretical results, the numerical simulations, and the experimental results. Moreover, a typical fuselage section of aircraft during a crash landing was considered and the fuselage section with the 3D NPR structure presented in this work showed better energy absorption capacity than the one without the 3D NPR structure.