Biomechanical and functional properties of tissue engineered vascular grafts must be similar to those observed in native vessels. This supposes a complete mechanical and structural characterization of the blood vessels. To this end, static and dynamic mechanical tests performed in the sheep thoracic and abdominal aorta and the cava vein were contrasted with histological quantification of their main constituents: elastin, collagen and muscle cells. Our results demonstrate that in order to obtain adequate engineered vascular grafts, the absolute amount of collagen fibers, the collagen/elastin ratio, the amount of muscle cells and the muscle cells/elastic fibers ratio are necessary to be determined in order to ensure adequate elastic modulus capable of resisting high stretches, an adequate elastic modulus at low and normal stretch values, the correct viscous energy dissipation, and a good dissipation factor and buffering function, respectively.