3D bioprinting techniques have shown great promise in various fields of tissue engineering and regenerative medicine. Yet, creating a tissue construct that faithfully represents the tightly regulated composition, microenvironment, and function of native tissues is still challenging. Among various factors, biomechanics of bioprinting processes play fundamental roles in determining the ultimate outcome of manufactured constructs. This review provides a comprehensive and detailed overview on various biomechanical factors involved in tissue bioprinting, including those involved in pre, during, and post printing procedures. In preprinting processes, factors including viscosity, osmotic pressure, and injectability are reviewed and their influence on cell behavior during the bioink preparation is discussed, providing a basic guidance for the selection and optimization of bioinks. In during bioprinting processes, we review the key characteristics that determine the success of tissue manufacturing, including the rheological properties and surface tension of the bioink, printing flow rate control, process-induced mechanical forces, and the in situ cross-linking mechanisms. Advanced bioprinting techniques, including embedded and multi-material printing, are explored. For post printing steps, general techniques and equipment that are used for characterizing the biomechanical properties of printed tissue constructs are reviewed. Furthermore, the biomechanical interactions between printed constructs and various tissue/cell types are elaborated for both in vitro and in vivo applications. The review is concluded with an outlook regarding the significance of biomechanical processes in tissue bioprinting, presenting future directions to address some of the key challenges faced by the bioprinting community.