Myocardial infarction (MI) causes massive heart muscle death and remains a leading cause of death in the world. Cardiac tissue engineering aims to replace the infarcted tissues with functional engineered heart muscles or revitalize the infarcted heart by delivering cells, bioactive factors, and/or biomaterials. One major challenge of cardiac tissue engineering and regeneration is the establishment of functional perfusion and structure to achieve timely angiogenesis and effective vascularization, which are essential to the survival of thick implants and the integration of repaired tissue with host heart. In this paper, we review four major approaches to promoting angiogenesis and vascularization in cardiac tissue engineering and regeneration: delivery of pro-angiogenic factors/molecules, direct cell implantation/cell sheet grafting, fabrication of prevascularized cardiac constructs, and the use of bioreactors to promote angiogenesis and vascularization. We further provide a detailed review and discussion on the early perfusion design in nature-derived biomaterials, synthetic biodegradable polymers, tissue-derived acellular scaffolds/whole hearts, and hydrogel derived from extracellular matrix. A better understanding of the current approaches and their advantages, limitations, and hurdles could be useful for developing better materials for future clinical applications.