Additive manufacturing (AM) techniques are becoming the approaches of choice for the construction of scaffolds in tissue engineering. However, the development of 3D printing in this field brings unique challenges, which must be accounted for in the design of experiments. The common printing process parameters must be considered as important factors in the design and quality of final 3D-printed products. In this work, we study the influence of some parameters in the design and fabrication of PCL scaffolds, such as the number and orientation of layers, but also others of “hidden” importance, such as the cooling down rate while printing, or the position of the starting point in each layer. These factors can have an important impact oin the final porosity and mechanical performance of the scaffolds. A pure polycaprolactone filament was used. Three different configurations were selected for the design of the internal structure of the scaffolds: a solid one with alternate layers (solid) (0°, 90°), a porous one with 30% infill and alternate layers (ALT) (0°, 90°) and a non-alternated configuration consisting in printing three piled layers before changing the orientation (n-ALT) (0°, 0°, 0°, 90°, 90°, 90°). The nozzle temperature was set to 172 °C for printing and the build plate to 40 °C. Strand diameters of 361 ± 26 µm for room temperature cooling down and of 290 ± 30 µm for forced cooling down, were obtained. A compression elastic modulus of 2.12 ± 0.31 MPa for n-ALT and 8.58 ± 0.14 MPa for ALT scaffolds were obtained. The cooling down rate has been observed as an important parameter for the final characteristics of the scaffold.