A hood covers an internal combustion engine on the rear of a grass mowing machine. The hood has a top surface with a plurality of air intake openings recessed below the top surface, and an integral air duct. The integral air duct extends over the engine between the air intake openings and an air inlet side of the radiator and/or oil cooler adjacent the rear of the grass mowing machine. The air intake openings closest to the air inlet side of the radiator and/or oil cooler are dimensioned to limit the air intake velocity and prevent buildup of debris on screens or grilles over the openings.