This paper presents the design and construction of a mechanical positioning system, which is part of a mechatronic prototype that integrates the areas of electronics, mechanics, control and computing, forming a synergy leading to create an automated machine that meets the technical requirements and requirements for a 3D printer, using a rapid prototyping technology (Fused Deposition Modeling: FDM), to produce a part designed on a CAD software, through the positioning and the action of a plastic extruder tool in three dimensions. The proposed methodology provides, step by step, the design and construction of the mechanical system of three-dimensional Cartesian positioning, which takes into account the variables and processes necessary for optimal response of the machine, with regard to surface finishing, printing time, emergency stops, and so forth.