The building construction sector is undergoing one of the most profound transformations towards the digital transition of production. In recent decades, the advent of a novel technology for the 3D printing of clay opened up new sustainable possibilities in construction. Some architectural applications of 3D-printed clay bricks with simple internal configurations are being developed around the world. On the other hand, the full potential of 3D-printed bricks for building production is still unknown. Scientific studies about the design and printability of 3D-printed bricks exploiting complex internal geometries are completely missing in the related literature. This paper explores the new boundaries of 3D-printed clay bricks realized with a sustainable extrusion-based 3D clay printing process by proposing a novel conception, design, and analysis. In particular, the proposed methodological approach includes: (i) conception and design; (ii) parametric modeling; (iii) simulation of printability; and (iv) prototyping. The new design and conception aim to fully exploit the potential of 3D printing to realize complex internal geometry in a 3D-printed brick. To this aim, the research investigates the printability of internal configuration generated by using geometries with well-known remarkable mechanical properties, such as periodic minimal surfaces. In conclusion, the results are validated by a wide prototyping campaign.