The 3D bioprinting technique has made enormous progress in tissue engineering, regenerative medicine and research into diseases such as cancer. Apart from individual cells, a collection of cells, such as organoids, can be printed in combination with various hydrogels. It can be hypothesized that 3D bioprinting will even become a promising tool for mechanobiological analyses of cells, organoids and their matrix environments in highly defined and precisely structured 3D environments, in which the mechanical properties of the cell environment can be individually adjusted. Mechanical obstacles or bead markers can be integrated into bioprinted samples to analyze mechanical deformations and forces within these bioprinted constructs, such as 3D organoids, and to perform biophysical analysis in complex 3D systems, which are still not standard techniques. The review highlights the advances of 3D and 4D printing technologies in integrating mechanobiological cues so that the next step will be a detailed analysis of key future biophysical research directions in organoid generation for the development of disease model systems, tissue regeneration and drug testing from a biophysical perspective. Finally, the review highlights the combination of bioprinted hydrogels, such as pure natural or synthetic hydrogels and mixtures, with organoids, organoid-cell co-cultures, organ-on-a-chip systems and organoid-organ-on-a chip combinations and introduces the use of assembloids to determine the mutual interactions of different cell types and cell-matrix interferences in specific biological and mechanical environments.