The work concerns a three-point bending test of beams made of plywood, high density fibre boards, cardboard, and wood-epoxy mass. The goal of the investigation was to determine the effect of thickness and type of wood-based facings on stiffness, strength, ability to absorb, and dissipate the energy of sandwich beams with an auxetic core. The cognitive goal of the work was to demonstrate the possibility of using recycled materials for facings and cores instead of popular wood composites. Experimental studies and numerical calculations were performed on correctly calibrated models. Experimental studies have shown that the beams with HDF facings (E = 1528 MPa, MOR = 12.61 MPa) and plywood facings (E = 1248–1395 MPa, MOR = 8.34–10.40 MPa) have the most favourable mechanical properties. Beams with plywood facings also have a good ability to absorb energy (1.380–1.746 J), but, in this respect, the beams manufactured of HDF (2.223 J) exhibited better capacity. The use of an auxetic core and facings of plywood and cardboard significantly reduces the amount of dissipated energy (0.0093 J, 0.0067 J). Therefore, this type of structures can be used for modeling beams carrying high deflections.