Interest in recapitulating in vivo phenomena in vitro using organ-on-a-chip technology has grown rapidly and with it, attention to the types of fluid flow experienced in the body has followed suit. These platforms offer distinct advantages over in vivo models with regards to human relevance, cost, and control of inputs (e.g., controlled manipulation of biomechanical cues from fluid perfusion). Given the critical role biophysical forces play in several tissues and organs, it is therefore imperative that engineered in vitro platforms capture the complex, unique flow profiles experienced in the body that are intimately tied with organ function. In this review, we outline the complex and unique flow regimes experienced by three different organ systems: blood vasculature, lymphatic vasculature, and the intestinal system. We highlight current state-of-the-art platforms that strive to replicate physiological flows within engineered tissues while introducing potential limitations in current approaches.