This article is devoted to derivation of new integral representations for the main thermoelastic Green's functions (MTGFs), based on the presentation of solutions of respective Lamé elliptic differential equations via Green's functions for the Poisson equation (GFPEs). The newly derived integral representations in Cartesian coordinates permitted the proof of a theorem about constructive formulas for MTGFs expressed in terms of respective GFPEs. The thermoelastic displacements are generated by a unitary heat source, applied in an arbitrary inner point of a generalized boundary values problem (BVP) of thermoelasticity for an octant at different homogeneous mechanical and thermal boundary conditions, prescribed on its marginal quadrants. According to the constructive formulas obtained, the derivation of MTGFs for about 20 BVPs for a plane, a half-plane, a quadrant, a space, a quarter-space, and an octant may be obtained by changing the respective well-known GFPEs. All results obtained are in terms of elementary functions with many examples of their validation. Two new MTGFs for quarter-space and octant, together with some of their graphical computer evaluations, are also included. The main advantages of the proposed approach in comparison with the GΘ convolution method for MTGFs constructing are: First, it is not necessary to derive the functions of influence of a unit concentrated force onto elastic volume dilatation - Θ⁽ⁱ⁾. Second, it is not necessary to calculate an integral of the product of the volume dilatation and Green's function in heat conduction. By using the proposed approach it is possible to extend obtained results for Cartesian domains onto areas of any orthogonal system of coordinates.