A nozzle segment for a gas turbine engine may generally include an airfoil having an exterior surface defining a pressure side and a suction side extending between leading and trailing edges. The airfoil may define an open internal volume within its interior for receiving a cooling medium. The open internal volume may include a primary internal cavity and a plurality of internal cooling channels in flow communication with the primary internal cavity. The primary internal cavity may extend within the interior of the airfoil from a location adjacent to the leading edge to a forward end of each of the internal cooling channels. The internal cooling channels may extend within the interior of the airfoil from the primary internal cavity towards the trailing edge. In addition, the internal cooling channels may be spaced apart radially by a plurality of ribs extending within the interior of the airfoil.