Dynamic human organs-on-chips are microfluidic devices capable of reproducing the physiology of human organs, accurately simulating both normal and disease states. These microphysiological systems offer more predictive and ethical alternatives both to animal-based preclinical drug studies and to cell-based in vitro diagnostics. But multiple organs-on-chips must be connected together in order to emulate the full human body. Therefore, new and more complex organs-on-chips that are more representative and versatile need to be designed and developed. The most salient challenge lies in their fabrication. Current prototypes rely on traditional microfluidic and microelectromechanical system methods that are too time-consuming for rapid prototyping, troubleshooting, and customization. Newer three-dimensional bioprinting technologies can help facilitate fabrication and deployment of next generation organs-on-chips. This review of the literature provides background on the clinical utility of dynamic human organs-on-chips. It lays out their design and fabrication requirements in the context of state of the art bioprinting technologies. It also proposes development pathways for bioprinted organs-on-chips to engineer more predictive designs. Finally, regulatory considerations and the impact of future bioprinting approaches on organ-on-chips are discussed with emphasis on enabling an era of personalized medicine.