A platform cooling arrangement in a turbine rotor blade having a platform positioned between an airfoil and a root. The rotor blade, along a side that coincides with a pressure side of the airfoil, includes a pressure side of the platform includes a topside extending from an airfoil base to a pressure side slashface. The platform cooling arrangement includes: a main plenum residing just inboard of the topside in the pressure side of the platform, the main plenum extending through the platform from an upstream end having an aft position to a downstream end having a forward position; and cooling apertures. Near the upstream end, the main plenum includes an aft switchback, and, between the aft switchback and the downstream end, a forward arc. Each of the cooling apertures extends from the main plenum to a port formed on the pressure side slashface.