Bioprinting has emerged as a versatile biofabrication approach for creating tissue engineered organ constructs. These constructs have potential use as organ replacements for implantation in patients, and also, when created on a smaller size scale as model "organoids" that can be used in in vitro systems for drug and toxicology screening. Despite development of a wide variety of bioprinting devices, application of bioprinting technology can be limited by the availability of materials that both expedite bioprinting procedures and support cell viability and function by providing tissue-specific cues. Here we describe a versatile hyaluronic acid (HA) and gelatin-based hydrogel system comprised of a multi-crosslinker, 2-stage crosslinking protocol, which can provide tissue specific biochemical signals and mimic the mechanical properties of in vivo tissues. Biochemical factors are provided by incorporating tissue-derived extracellular matrix materials, which include potent growth factors. Tissue mechanical properties are controlled combinations of PEG-based crosslinkers with varying molecular weights, geometries (linear or multi-arm), and functional groups to yield extrudable bioinks and final construct shear stiffness values over a wide range (100 Pa to 20 kPa). Using these parameters, hydrogel bioinks were used to bioprint primary liver spheroids in a liver-specific bioink to create in vitro liver constructs with high cell viability and measurable functional albumin and urea output. This methodology provides a general framework that can be adapted for future customization of hydrogels for biofabrication of a wide range of tissue construct types.