Three-dimensional (3D) bioprinting is a fabrication method with many biomedical applications, particularly within tissue engineering. The use of freezing during 3D bioprinting, aka “3D cryoprinting,” can be utilized to create micopores within tissue-engineered scaffolds to enhance cell proliferation. When used with alginate bio-inks, this type of 3D cryoprinting requires three steps: 3D printing, crosslinking, and freezing. This study investigated the influence of crosslinking order and cooling rate on the microstructure and mechanical properties of sodium alginate scaffolds. We designed and built a novel modular 3D printer in order to study the effects of these steps separately and to address many of the manufacturing issues associated with 3D cryoprinting. With the modular 3D printer, 3D printing, crosslinking, and freezing were conducted on separate modules yet remain part of a continuous manufacturing process. Crosslinking before the freezing step produced highly interconnected and directional pores, which are ideal for promoting cell growth. By controlling the cooling rate, it was possible to produce pores with diameters from a range of 5 μm to 40 μm. Tensile and firmness testing found that the use of freezing does not decrease the tensile strength of the printed objects, though there was a significant loss in firmness for strands with larger pores.