Three-dimensional (3D) bioprinting has emerged with potential for creating functional 3D tissues with customized geometries. However, the limited availability of bioinks capable of printing 3D structures with both high-shape fidelity and desired biological environments for encapsulated cells remains a key challenge. Here, we present a 3D bioprinting approach that uses universal fugitive network bioinks prepared by loading cells and hydrogel precursors (bioink base materials) into a 3D printable fugitive carrier. This approach constructs 3D structures of cell-encapsulated hydrogels by printing 3D structures using fugitive network bioinks, followed by cross-linking printed structures and removing the carrier from them. The use of the fugitive carrier decouples the 3D printability of bioinks from the material properties of bioink base materials, making this approach readily applicable to a range of hydrogel systems. The decoupling also enables the design of bioinks for the biological functionality of the final printed constructs without compromising the 3D printability. We demonstrate the generalizable 3D printability by printing self-supporting 3D structures of various hydrogels, including conventionally non-3D printable hydrogels and those with a low polymer content. We conduct preprinting screening of bioink base materials through 3D cell culture to select bioinks with high cell compatibility. The selected bioinks produce 3D constructs of cell-encapsulated hydrogels with both high-shape fidelity and high cell viability and proliferation. The universal fugitive network bioink platform could significantly expand 3D printable bioinks with customizable biological functionalities and the adoption of 3D bioprinting in diverse research and applied settings.