We report the fabrication of a scaffold (hereafter referred to as AngioChip) that supports the assembly of parenchymal cells on a mechanically tunable matrix surrounding a perfusable, branched, three-dimensional microchannel network coated with endothelial cells. The design of AngioChip decouples the material choices for the engineered vessel network and for cell seeding in the parenchyma, enabling extensive remodelling while maintaining an open-vessel lumen. The incorporation of nanopores and micro-holes in the vessel walls enhances permeability, and permits intercellular crosstalk and extravasation of monocytes and endothelial cells on biomolecular stimulation. We also show that vascularized hepatic tissues and cardiac tissues engineered by using AngioChips process clinically relevant drugs delivered through the vasculature, and that millimetre-thick cardiac tissues can be engineered in a scalable manner. Moreover, we demonstrate that AngioChip cardiac tissues implanted with direct surgical anastomosis to the femoral vessels of rat hindlimbs establish immediate blood perfusion.