Porous poly-ether-ether-ketone (PEEK) scaffolds have a good application prospect in orthopedics because of their excellent biocompatibility, chemical stability, and permeability. However, the porous PEEK scaffolds fabricated by conventional manufacturing methods are restricted to the uncontrollable and weak mechanical properties. In this work, porous PEEK scaffolds with highly-tunable elastic modulus are designed using typical triply periodic minimal surface (TPMS) structures, Gyroid, Diamond, I-WP, and then printed by laser powder bed fusion (LPBF). The effects of unit cell type and volume fraction on the manufacturing deviation, stress distribution, and mechanical properties of LPBF-printed PEEK scaffolds are investigated. The results illustrate that the Gyroid scaffolds present the lowest volume fraction deviation, followed by the I-WP scaffolds. By adjusting the volume fraction, the elastic modulus of the TPMS scaffolds is in the range of 42.17–512.12 MPa, which is comparable with those of human trabecular bones. Using the finite element (FE) simulation method, the experimental mechanical properties of the TPMS scaffolds can be predicted. In summary, the Gyroid structure is the optimal choice for the design and LPBF fabrication of porous PEEK scaffolds due to the best manufacturing fidelity and relatively good mechanical properties.