Structures with triply periodic minimal surfaces (TPMS) have recently been shown to possess attractive mechanical properties such as high stiffness-to-density and strength-to-density ratios. In this study, we investigate the absorption coefficient of porous TPMS structures with a view towards their possible use as acoustic liners for reducing aircraft engine noise. Porous structures with controlled microstructures were fabricated for experimental testing using MATLAB and a Form 2 stereolithographic 3D printer. The sound absorption coefficient of the fabricated structures was measured using a two-microphone impedance tube setup and in accordance with ASTM E1050-12. The effect of surface topology on the acoustic properties was characterized by testing four different TPMS geometries. The gyroid geometry was then selected to study the effect of relative density on its acoustic properties. Finally, the effect of functionally-graded through-thickness relative density variation was studied. The obtained results show that TPMS structures offer a novel path towards the design of multifunctional structures with good mechanical and acoustical properties.