Bioprinting is a new technology, which arranges cells with high spatial resolution, but its potential to create models for viral infection studies has not yet been fully realized. The present study describes the optimization of a bioink composition for extrusion printing. The bioinks were biophysically characterized by rheological and electron micrographic measurements. Hydrogels consisting of alginate, gelatin and Matrigel were used to provide a scaffold for a 3D arrangement of human alveolar A549 cells. A blend containing 20% Matrigel provided the optimal conditions for spatial distribution and viability of the printed cells. Infection of the 3D model with a seasonal influenza A strain resulted in widespread distribution of the virus and a clustered infection pattern that is also observed in the natural lung but not in two-dimensional (2D) cell culture, which demonstrates the advantage of 3D printed constructs over conventional culture conditions. The bioink supported viral replication and proinflammatory interferon release of the infected cells. We consider our strategy to be paradigmatic for the generation of humanized 3D tissue models by bioprinting to study infections and develop new antiviral strategies.