By tailoring their microstructural features, microstructured materials can exhibit widely tunable effective properties that are unreachable or even beyond those of the base materials, such as negative Poisson’s ratio and negative thermal expansion coefficient. Strut-based lattice structures are typical examples. If more than one base material is introduced in a lattice structure, its effective properties rely on the base material properties as well. To this end, a much wider design space is attained. In practice, the design and application of microstructured materials are two-fold tasks: forward prediction and inverse design. The former tackles the prediction of effective properties of a specified microstructure. The latter is aimed at designing a microstructure with specified target effective properties. In this contribution, we introduce a few novel bimaterial strut-based lattice structures. They can exhibit negative Poisson’s ratios and/or negative thermal expansion coefficients. We consider their inverse design by a data-driven method. To this end, we exploit the computational homogenization method to evaluate the effective thermoelastic properties of numerous structure designs with varying structural features, which results in a dataset consisting of structural features paired with the corresponding effective properties. For each type of lattice structure, we construct two artificial neural network (ANN) surrogate models for the forward prediction and inverse design. The ANN models are trained and verified with the dataset. Subsequently, the performance of these data-driven surrogate models is illustrated by typical forward and inverse design tasks. The data-driven surrogate models will facilitate the application of these novel bimaterial lattice structures in different engineering scenarios for structural and/or functional purposes. The introduced data-driven method can be extended to realize the inverse design of other types of microstructured materials.