Improving the post-processing of metallic porous tissue scaffolds is an essential step to create a new generation of superelastic implants for the replacement of damaged bone tissue. In this study, the dynamic chemical etching technique is applied to improve the permeability and to optimize the porous structure of Ti-Nb-Zr scaffolds fabricated by the powder metallurgy-based space holder technique. The etched scaffolds are characterized in terms of their porous structure geometry, permeability, and mechanical properties. It is shown that an increase in porosity from 49% to 54% during the etching is mainly due to an increase in the number of 100 to 800 μm-diameter pores, from 30% to 50% of them measuring from 100 to 300 μm in size. These changes in the porous structure lead to a significant increase of its permeability, i.e., from (0.1–15) × 10−11 m2 before etching to (44–91) × 10−11 m2, after etching; these permeability ranges corresponding to those of bone tissues. Furthermore, the etched scaffolds show systematically higher yield compressive stresses as compared to the as-sintered scaffolds of equivalent porosities. Finally, the highly permeable etched Ti-Nb-Zr scaffolds with a porosity varying from 40% to 60% exhibit an apparent Young’s modulus ranging from 8.6 to 1.9 GPa and an ultimate compressive strength from 650 to 190 MPa, which can be considered as a promising balance of properties for the potential use of these scaffolds as bone implants.