Microchannel heat sinks are capable of removing dense heat loads from high-power electronic devices with low thermal resistance, but suffer from high pressure drops due to the small channel dimensions. Features that reduce the pressure drop, such as manifolds, increase fabrication complexity and are constrained by traditional subtractive manufacturing approaches. Additive manufacturing technologies offer improved design freedom and reduced geometric restrictions, expanding the types of features that can be produced and integrated into a heat sink. In this work, a novel permeable membrane microchannel (PMM) heat sink geometry is proposed and fabricated using direct metal laser sintering (DMLS) of an aluminum alloy (AlSi10Mg). In this PMM design, the cooling fluid is forced through thin, porous walls that act as both conducting fins and membranes that allow flow through their fine internal flow features for efficient heat exchange. The design leverages the ability of this fabrication process to incorporate complex, arbitrarily curved structures having internal porosity to enhance heat transfer and reduce pressure drop across the heat sink. The PMM heat sink geometry is benchmarked against a low-pressure-drop manifold microchannel (MMC) heat sink. A reduced-order model is used to explore the relative performance trends between the designs. Both heat sinks are experimentally characterized at flow rates of 50-500 mL/min using deionized water as the working fluid. At a constant pumping power of 0.018 W, the permeable membrane microchannel design offers both lower thermal resistance (17% reduction) and lower pressure drop (28% reduction) compared to the manifold microchannel heat sink.