•A class of novel 3D auxetic lattice structures are developed based on the rotating rigid mechanism.•Analytical models for predicting the elastic properties of the new structure are developed.•The structures have been shown excellent design flexibility with respect to Poisson’s ratio. This paper introduces a new methodology for generating three-dimensional (3D) negative Poisson’s ratio behavior with lattice representations of the rotating rigid mechanism as a starting point. Based on it, a class of new 3D auxetic lattice structures is proposed. The elastic properties of a representative 3D auxetic lattice structure, including the homogenized Young’s modulus and Poisson’s ratio along the three principal axes, are systematically investigated in a combination of analytical predictions, numerical simulations and experimental tests. Moreover, effects of the structural geometrical parameters and specimen size on the elastic properties as well as structural nonlinear mechanical responses along the principal axes are carefully discussed. Different from most traditional 3D auxetic materials that are only capable of achieving negative or positive Poisson's ratio, our results suggest that the homogenized Poisson’s ratio effect of the proposed 3D lattice structure along all principal axes can be tuned from positive to negative in a wide range. The excellent design flexibility would help to expand and hasten the adoption of the new 3D auxetic lattice structures in engineering applications.