Advancements in bioinks and three-dimensional (3D) printing and bioprinting have significantly advanced cardiovascular tissue engineering by enabling the fabrication of biomimetic cardiac and vascular constructs. Traditional 3D printing has contributed to the development of acellular scaffolds, vascular grafts, and patient-specific cardiovascular models that support surgical planning and biomedical applications. In contrast, 3D bioprinting has emerged as a transformative biofabrication technology that allows for the spatially controlled deposition of living cells and biomaterials to construct functional tissues in vitro. Bioinks—derived from natural biomaterials such as collagen and decellularized matrix, synthetic polymers such as polyethylene glycol (PEG) and polycaprolactone (PCL), or hybrid combinations—have been engineered to replicate extracellular environments while offering tunable mechanical properties. These formulations ensure biocompatibility, appropriate mechanical strength, and high printing fidelity, thereby maintaining cell viability, structural integrity, and precise architectural resolution in the printed constructs. Advanced bioprinting modalities, including extrusion-based bioprinting (such as the FRESH technique), droplet/inkjet bioprinting, digital light processing (DLP), two-photon polymerization (TPP), and melt electrowriting (MEW), enable the fabrication of complex cardiovascular structures such as vascular patches, ventricle-like heart pumps, and perfusable vascular networks, demonstrating the feasibility of constructing functional cardiac tissues in vitro. This review highlights the respective strengths of these technologies—for example, extrusion’s ability to print high-cell-density bioinks and MEW’s ultrafine fiber resolution—as well as their limitations, including shear-induced cell stress in extrusion and limited throughput in TPP. The integration of optimized bioink formulations with appropriate printing and bioprinting platforms has significantly enhanced the replication of native cardiac and vascular architectures, thereby advancing the functional maturation of engineered cardiovascular constructs.