The demand for robust yet lightweight materials has exponentially increased in several engineering applications. Additive manufacturing and 3D printing technology have the ability to meet this demand at a fraction of the cost compared with traditional manufacturing techniques. By using the fused deposition modeling (FDM) or fused filament fabrication (FFF) technique, objects can be 3D-printed with complex designs and patterns using cost-effective, biodegradable, and sustainable thermoplastic polymer filaments such as polylactic acid (PLA). This study aims to provide results to guide users in selecting the optimal printing and testing parameters for additively manufactured/3D-printed components. This study was designed using the Taguchi method and grey relational analysis. Compressive test results on nine similarly patterned samples suggest that cuboid gyroid-structured samples perform the best under compression and retain more mechanical strength than the other tested triply periodic minimal surface (TPMS) structures. A printing speed of 40 mm/s, relative density of 60%, and cell size of 3.17 mm were the best choice of input parameters within the tested ranges to provide the optimal performance of a sample that experiences greater force or energy to compress until failure. The ninth experiment on the above-mentioned conditions improved the yield strength by 16.9%, the compression modulus by 34.8%, and energy absorption by 29.5% when compared with the second-best performance, which was obtained in the third experiment.