Despite the significant technological advancement in tissue engineering, challenges still exist towards the development of complex and fully functional tissue constructs that mimic their natural counterparts. To address these challenges, bioprinting has emerged as an enabling technology to create highly organized three-dimensional (3D) vascular networks within engineered tissue constructs to promote the transport of oxygen, nutrients, and waste products, which can hardly be realized using conventional microfabrication techniques. Here, we report the development of a versatile 3D bioprinting strategy that employs biomimetic biomaterials and an advanced extrusion system to deposit perfusable vascular structures with highly ordered arrangements in a single-step process. In particular, a specially designed cell-responsive bioink consisting of gelatin methacryloyl (GelMA), sodium alginate, and 4-arm poly(ethylene glycol)-tetra-acrylate (PEGTA) was used in combination with a multilayered coaxial extrusion system to achieve direct 3D bioprinting. This blend bioink could be first ionically crosslinked by calcium ions followed by covalent photocrosslinking of GelMA and PEGTA to form stable constructs. The rheological properties of the bioink and the mechanical strengths of the resulting constructs were tuned by the introduction of PEGTA, which facilitated the precise deposition of complex multilayered 3D perfusable hollow tubes. This blend bioink also displayed favorable biological characteristics that supported the spreading and proliferation of encapsulated endothelial and stem cells in the bioprinted constructs, leading to the formation of biologically relevant, highly organized, perfusable vessels. These characteristics make this novel 3D bioprinting technique superior to conventional microfabrication or sacrificial templating approaches for fabrication of the perfusable vasculature. We envision that our advanced bioprinting technology and bioink formulation may also have significant potentials in engineering large-scale vascularized tissue constructs towards applications in organ transplantation and repair.