Respiratory diseases, e.g., idiopathic pulmonary fibrosis, are increasing the expanding healthcare burden globally. To develop effective treatments, lung tissue engineering (LungTE) provides a promising approach. The mechanical characteristics of human lungs change from elastic and compliant alveolar regions to stiffer and stronger bronchial and pleural regions. This progressive change in mechanical microenvironment plays an important role in lung recovery from disease to the healthy state after medical intervention. In scaffold-based tissue engineering (TE), TE scaffolds provide a conducive microenvironment for cells and for new tissue formation. Compared to other technologies, many 3D printing technologies have distinctive advantages of producing complex TE scaffolds with gradients (structural, mechanical, chemical, and/or biological). Current research on LungTE mainly uses hydrogels with non-variable mechanical properties. In this study, new hydrogel printing inks comprising sodium alginate, carboxylated cellulose nanofibers and methacrylic acid-modified silk fibroin (MeSF) were formulated and assessed for extrusion-based 3D printing into novel LungTE scaffolds. They could have controlled dual-crosslinking, thus enabling 3D printing of LungTE scaffolds with programmable mechanical gradients. A physically crosslinked primary network, controlled by polymer chain entanglement and ionic interaction, was formed by sodium alginate and cellulose nanofibers, while MeSF enabled formation of a second covalent network via post-printing photopolymerization. Concentrations of the three components in new hydrogels could be changed, adding another factor for controlling mechanical properties of scaffolds. Rheological analysis was conducted for hydrogel inks to achieve good printability. Printing parameters were optimized. Even for intricate alveolar-like porous designs, hydrogel scaffolds showed no collapse or filament fusion after 3D printing and retained structural integrity during mechanical deformation. Dual-crosslinking enabled high scaffold fidelity and spatially adjustable mechanical property, which could lead to creation of mechanically distinct peripheral, alveolar, and bronchial sections within a single 3D printed scaffold. An in vitro biological study revealed good biocompatibility of printed scaffolds.