A theory of thermoelastic polynomials that determine fundamental solutions of a central equation of linear thermoelasticity with finite wave speeds is developed. The polynomials define series solution of the Neumann type for an integro-differential equation on the time axis associated with a method of separation of variables for the central equation. A number of recurrence relations defining basic properties of the thermoelastic polynomials are presented. In particular, it is shown that the polynomials can be identified with an element of the null space of a linear ordinary differential operator.