Disclosed are rigid or semi-rigid multilayer structures that provide high oxygen permeability values comprising metallocene polyethylene, ethylene/vinyl acetate copolymers or highly neutralized blends of ethylene acid copolymers, organic acid salts and optionally one or more ethylene-containing polymers or silicon-containing polymers. Such structures have improved oxygen permeability and are useful in packaging food products such as case ready meat, fish, sausage, fresh produce, and the like that require breathable packages.