The aim of this study is to derive new constructive formulas and analytical expressions for Green’s functions (GFs) to 3D generalized boundary value problem (BVP) for an unbounded parallelepiped under a point heat source. These results were obtained using the developed harmonic integral representation method. On the base of derived constructive formulas it is possible to obtain analytical expressions for thermal stresses GFs to 16 BVPs for unbounded parallelepiped. An example of such kind is presented for a spatial BVP, GFs of which are presented in the form of the sum of elementary functions and double infinite series, containing products between exponential and trigonometric functions. An integration formula for thermal stresses, caused by the thermal data, distributed on the boundary strips at homogeneous locally mixed mechanical boundary conditions was also derived. The main difficulty to obtain these results was calculating an integral of the product between two GFs for Poisson’s equation. This integral taken on the base of the earlier established statement that main thermoelastic displacement Green’s functions (MTDGFs) satisfy the boundary conditions: (a) homogeneous mechanical conditions with respect to points of findings MTDGFs and (b) homogeneous thermal conditions with respect to points of the application of the heat source.