The scientific literature regarding additive manufacturing, mainly the material extrusion method, suggests that the mechanical characteristics of the parts obtained by this technology depend on a number of the input factors specific to the printing process, such as printing temperature, printing trajectory, layer height, etc., and also on the post-process operations for parts, which, unfortunately, requires supplementary setups, equipment, and multiple steps that raise the overall costs. Therefore, this paper aims to investigate the influence of the printing direction, the thickness of the deposited material layer, and the temperature of the previously deposited material layer on the part tensile strength, hardness by means of Shore D and Martens hardness, and surface finish by using an in-process annealing method. A Taguchi L9 DOE plan was developed for this purpose, where the test specimens, with dimensions according to ISO 527-2 type B, were analysed. The results showed that the presented in-process treatment method is possible and could lead to sustainable and cost-effective manufacturing processes. The varied input factors influenced all the studied parameters. Tensile strength tended to increase, up to 12.5%, when the in-process heat treatment was applied, showed a positive linear variation with nozzle diameter, and presented considerable variations with the printing direction. Shore D and Martens hardness had similar variations, and it could be observed that by applying the mentioned in-process heat treatment, the overall values tended to decrease. Printing direction had a negligible impact on the additively manufactured parts’ hardness. At the same time, the nozzle diameter presented considerable variations, up to 36% for Martens hardness and 4% for Shore D, when higher diameter nozzles were used. The ANOVA analysis highlighted that the statistically significant factors were the nozzle diameter for the part’s hardness and the printing direction for the tensile strength.