<p>This paper is concerned with a linear theory of porous thermoelastic materials in which the second temperature gradient is included in the classical set of independent constitutive variables. The paper is based on the theory of microstretch thermoelastic solids, as well as on Green-Naghdi thermomechanics. We use thermal displacement and an entropy production inequality. The introduction of the entropy flux tensor allows the constitutive equations to depend on the second gradient of temperature. We first present the basic equations of the theory as well as the boundary conditions for this class of non-simple materials. We then study the case of isotropic and homogeneous materials and present a general solution of the field equations similar to that obtained by Mindlin in strain gradient elasticity. In the context of anisotropic solids we discuss the uniqueness question appropriate to the fundamental initial-boundary-value problems. The continuous dependence of solutions on initial data and body loads is established. The Mindlin-type solution is used to study the deformation produced by a concentrated heat source in a body occupying an unbounded region.</p>