A cross-flow filtration device which separates a feed stock into filtrate and retentate, including a structure of porous material which defines a plurality of passageways extending longitudinally from the feed end of the structure to a retentate end, and a number of filtrate conduits within the structure for carrying filtrate from within the structure toward a filtrate collection zone. The filtrate conduits provide paths of lower flow resistance than that of alternative flow paths through the porous material, and the structure is constructed such that the filtrate conduits are distributed among the passageways to provide low pressure drop flow paths for filtrate flow from the passageways through the porous material to nearby filtrate conduits. The cross-flow filtration device can also be utilized as a membrane support for a device for cross-flow microfiltration, ultrafiltration, reverse osmosis, gas separations or pervaporation. Also disclosed is a method of making such a cross-flow filtration device.