3D printing is a technology that allows the fabrication of structures with arbitrary geometries and heterogeneous material properties. The application of this technology to biological structures that match the complexity of native tissue is of great interest to researchers. This mini-review highlights the current progress of 3D printing for fabricating artificial tissues of the cardiovascular system, specifically the myocardium, heart valves, and coronary arteries. In addition, how 3D printed sensors and actuators can play a role in tissue engineering is discussed. To date, all the work with building 3D cardiac tissues have been proof-of-principle demonstrations, and in most cases, yielded products less effective than other traditional tissue engineering strategies. However, this technology is in its infancy and therefore there is much promise that through collaboration between biologists, engineers and material scientists, 3D bioprinting can make a significant impact on the field of cardiovascular tissue engineering.