Cancer is a severe threat to human life and health and represents the main cause of death globally. Drug therapy is one of the primary means of treating cancer; however, most anticancer medications do not proceed beyond preclinical testing because the conditions of actual human tumors are not effectively mimicked by traditional tumor models. Hence, bionic in vitro tumor models must be developed to screen for anticancer drugs. Three-dimensional (3D) bioprinting technology can produce structures with built-in spatial and chemical complexity and models with accurately controlled structures, a homogeneous size and morphology, less variation across batches, and a more realistic tumor microenvironment (TME). This technology can also rapidly produce such models for high-throughput anticancer medication testing. This review describes 3D bioprinting methods, the use of bioinks in tumor models, and in vitro tumor model design strategies for building complex tumor microenvironment features using biological 3D printing technology. Moreover, the application of 3D bioprinting in vitro tumor models in drug screening is also discussed.