Honeycomb structures display extraordinary stiffness-to-weight ratio when loaded in the out-of-plane direction. When realized using thermoplastic polyurethane (TPU), the structures offer the potential for repeatable and high specific energy absorption. Varying the cell size and wall thickness of TPU honeycombs facilitates changes in stiffness magnitude, though affords only modest capacity to alter the shape of the stress-strain curve. 3D printing facilitates advanced design exploration, beyond that of straight walls. Origami fold patterns have demonstrated the ability to influence the buckling behavior of tubular structures. Here we demonstrate the incorporation of origami folds into square honeycombs. The fold parameters facilitate significant tailoring of the stress-strain curve, allowing a range of profiles from quasi-rectangular to quasi-linear to be achieved; such structures can find applications in situation-specific energy absorption scenarios. Display omitted •A method was developed to introduce origami fold patterns into square honeycombs.•The origami honeycombs were realized via 3D printing in thermoplastic polyurethane.•The crushing behavior of the origami honeycombs was studied via finite element analysis and experimental compression testing.•Varying the origami fold parameters allows significant tailoring of the honeycomb stress-strain response.•Absorption efficiencies as high as 0.49 were experimentally demonstrated, which rivals that of rigid polyurethane foams.