Cardiovascular tissue constructs provide unique design requirements due to their functional responses to substrate mechanical properties and cyclic stretching behavior of cardiac tissue that requires the use of durable elastic materials. Given the diversity of polyester synthesis approaches, an opportunity exists to develop a new class of biocompatible, elastic, and immunomodulatory cardiovascular polymers. Furthermore, elastomeric polyester materials have the capability to provide tailored biomechanical synergy with native tissue and hence reduce inflammatory response in vivo and better support tissue maturation in vitro. In this review, we highlight underlying chemistry and design strategies of polyester elastomers optimized for cardiac tissue scaffolds. The major advantages of these materials such as their tunable elasticity, desirable biodegradation, and potential for incorporation of bioactive compounds are further expanded. Unique fabrication methods using polyester materials such as micromolding, 3D stamping, electrospinning, laser ablation, and 3D printing are discussed. Moreover, applications of these biomaterials in cardiovascular organ-on-a-chip devices and patches are analyzed. Finally, we outline unaddressed challenges in the field that need further study to enable the impactful translation of soft polyesters to clinical applications.