The native tissues are complex structures consisting of different cell types, extracellular matrix materials, and biomolecules. Traditional tissue engineering strategies have not been able to fully reproduce biomimetic and heterogeneous tissue constructs because of the lack of appropriate biomaterials and technologies. However, recently developed three-dimensional bioprinting techniques can be leveraged to produce biomimetic and complex tissue structures. To achieve this, multicomponent bioinks composed of multiple biomaterials (natural, synthetic, or hybrid natural-synthetic biomaterials), different types of cells, and soluble factors have been developed. In addition, advanced bioprinting technologies have enabled us to print multimaterial bioinks with spatial and microscale resolution in a rapid and continuous manner, aiming to reproduce the complex architecture of the native tissues. This review highlights important advances in heterogeneous bioinks and bioprinting technologies to fabricate biomimetic tissue constructs. Opportunities and challenges to further accelerate this research area are also described.