2D cell culture and preclinical animal models have traditionally been implemented for investigating the underlying cellular mechanisms of human disease progression. However, the increasing significance of 3D vs. 2D cell culture has initiated a new era in cell culture research in which 3D in vitro models are emerging as a bridge between traditional 2D cell culture and in vivo animal models. Additive manufacturing (AM, also known as 3D printing), defined as the layer-by-layer fabrication of parts directed by digital information from a 3D computer-aided design file, offers the advantages of simultaneous rapid prototyping and biofunctionalization as well as the precise placement of cells and extracellular matrix with high resolution. In this review, we highlight recent advances in 3D printing of tissue engineered constructs that recapitulate the physical and cellular properties of the tissue microenvironment for investigating mechanisms of disease progression and for screening drugs.