A wide variety of experimental models including 2D cell cultures, model organisms, and 3D models have been developed to understand pathophysiological phenomena and assess the safety and efficacy of potential therapeutics. In this sense, 3D models are an intermediate between 2D cell cultures and animal models, as they adequately reproduce 3D microenvironments and human physiology while also being controllable and reproducible. Particularly, recent advances in 3D biomimicry models, which can produce complex cell structures, shapes, and arrangements, can more similarly reflect conditions than 2D cell culture. Based on this, 3D bioprinting technology, which enables to place the desired materials in the desired locations, has been introduced to fabricate tissue models with high structural similarity to the native tissues. Therefore, this review discusses the recent developments in this field and the key features of various types of 3D-bioprinted tissues, particularly those associated with blood vessels or highly vascularized organs, such as the heart, liver, and kidney. Moreover, this review also summarizes the current state of the three categories: (1) chemical substance treatment, (2) 3D bioprinting of lesions, and (3) recapitulation of tumor microenvironments (TME) of 3D bioprinting-based disease models according to their disease modeling approach. Finally, we propose the future directions of 3D bioprinting approaches for the creation of more advanced biomimetic 3D tissues, as well as the translation of 3D bioprinted tissue models to clinical applications.