A platform cooling arrangement in a turbine rotor blade having a platform that includes an interior cooling passage formed therein. The platform cooling arrangement may include: a main plenum residing just inboard of the planar topside and extending from an aft position to a forward position within one of the pressure side and the suction side of the platform, the main plenum having a longitudinal axis that is approximately parallel to the planar topside; a supply plenum that extends between the main plenum and the interior cooling passage; and a plurality of cooling apertures, each cooling aperture extending from one of the pressure side and the suction side slashface to a connection with the main plenum.