The increasing popularity of additive manufacturing technologies in the prototyping and building industry requires the application of novel, improved composite materials. In this paper, we propose the use of a novel 3D printing cement-based composite material with natural, granulated cork, and additional reinforcement using a continuous polyethylene interlayer net combined with polypropylene fibre reinforcement. Our assessment of different physical and mechanical properties of the used materials during the 3D printing process and after curing verified the applicability of the new composite. The composite exhibited orthotropic properties, and the compressive toughness in the direction of layer stacking was lower than that perpendicular to it, by 29.8% without net reinforcement, 42.6% with net reinforcement, and 42.9% with net reinforcement and an additional freeze–thaw test. The use of the polymer net as a continuous reinforcement led to decreased compressive toughness, lowering it on average by 38.5% for the stacking direction and 23.8% perpendicular to the stacking direction. However, the net reinforcement additionally lowered slumping and elephant’s foot effects. Moreover, the net reinforcement added residual strength, which allowed for the continuous use of the composite material after the failure of the brittle material. Data obtained during the process can be used for further development and improvement of 3D-printable building materials.