This article is concerned with a linear theory for elastic materials with inner structure, whose particles in addition to the classical displacement, possess microtemperatures. In the main part of the paper we restrict our attention to the one-dimensional problem. First, we prove the slow decay of solutions for the onedimensional problem of micromorphic elastic solids with the usual thermal effects. Then, we prove the exponential stability of the solutions when we consider the theory with microtemperatures. The anti-plane distributions of microtemperatures are considered later.