A continuum theory of pantographic lattices, based on second-grade elasticity, is presented. The proposed model is able to describe the mechanical behavior of a type of material structure made up of multiple layers of pantographic sheets connected with a third family of fibers. Thus, these materials are characterized by an orthogonal pattern of fibers that can bend, stretch and twist. Numerical experiments illustrate the predictive potential of the model when the material is subjected to different types of mechanical loads, including compression, torsion and two kinds of bending. Analyzing the material responses for these various tests makes it possible to reveal unusual deformation patterns characteristic of such “pantographic blocks.” Numerical simulations using the finite element method are intended to assist in designing an experimental program using 3D-printed specimens made of different materials.