Three-dimensional (3D) in vitro models, such as organ-on-a-chip platforms, are an emerging and effective technology that allows the replication of the function of tissues and organs, bridging the gap amid the conventional models based on planar cell cultures or animals and the complex human system. Hence, they have been increasingly used for biomedical research, such as drug discovery and personalized healthcare. A promising strategy for their fabrication is 3D printing, a layer-by-layer fabrication process that allows the construction of complex 3D structures. In contrast, 3D bioprinting, an evolving biofabrication method, focuses on the accurate deposition of hydrogel bioinks loaded with cells to construct tissue-engineered structures. The purpose of the present work is to conduct a systematic review (SR) of the published literature, according to the guidelines of the Preferred Reporting Items for Systematic Reviews and Meta-Analyses, providing a source of information on the evolution of organ-on-a-chip platforms obtained resorting to 3D printing and bioprinting techniques. In the literature search, PubMed, Scopus, and ScienceDirect databases were used, and two authors independently performed the search, study selection, and data extraction. The goal of this SR is to highlight the importance and advantages of using 3D printing techniques in obtaining organ-on-a-chip platforms, and also to identify potential gaps and future perspectives in this research field. Additionally, challenges in integrating sensors in organs-on-chip platforms are briefly investigated and discussed.