The present paper is devoted to formulations of constitutive equations for the non-linear Green–Naghdi type-III thermoelastic continuum consistent with the principle of thermodynamic (or thermomechanical) orthogonality. The principle of thermodynamic orthogonality proposed by Ziegler as a generalization of the Onsager linear irreversible thermodynamics states that the irreversible constituent parts of thermodynamic currents (velocities) are orthogonal to the convex dissipation potential level surface in the space of thermodynamic forces for any process of heat transport in a solid. The principle of the thermomechanical orthogonality takes its origin from the von Mises maximum principle of the perfect plasticity, where it provides existence of a yield surface, its convexity, and the associated flow rule. Non-linear constitutive laws of heat propagation as of type-III thermoelasticity complying with the principle of thermomechanical orthogonality are discussed. Important for applied thermoelasticity cases covered by type-III theory are studied: GNI/CTE – conventional thermoelasticity based on the Fourier heat conduction law and GNII – dissipationless hyperbolic thermoelasticity. In the latter case the internal entropy production equals zero for any heat transport process having the form of the undamped thermoelastic wave propagating at finite speed.