A co-extruded, multilayer film for packaging freshly cut produce consists of (1) an outer layer of an ethylene/propylene/4-8 C alpha-olefin terpolymer, (2) a middle layer containing at least one of (a) 0-100% of the terpolymer described in (1) and (b) 0-100% of a blend of (i) about 99.5% to about 95% of an olefin copolymer composition made by sequential polymerization in at least two stages and (ii) about 0.5% to about 5% of a low density polyethylene, wherein (a)+(b)=100%, and (3) an inner layer containing at least one of (a) 0-100% of a copolymer of ethylene and a C.sub.4 -C.sub.8 alpha-olefin made using a metallocene catalyst and having a density of 0.89 to 0.91 g/cm.sup.3, and (b) 0-100% of blend (b) described in layer (2), wherein (a)+(b)=100%. The thickness ratios of the layers and, when both components are present, the amount of each polymer in the blends of layers (2) and (3) can be varied to match the gas transmission rate of the produce to be packaged.