A turbine rotor includes a base and a plurality of blades. The base and the blades curve such that radially outward portions of the base and the blades extend in a direction with a greater component in a radial direction than in an axial direction. Radially central portions of the base and the blade extend in a direction with the two components being closer. Radially inner sections of the base and the blades extend in a direction with a greater component in the axial direction than in a radial direction. There is a cooling channel arrangement in the turbine rotor. The cooling channel arrangement includes impingement cooling for a nose and serpentine passages for cooling sections of the platform circumferentially intermediate the blades, and distinct serpentine passages for cooling the plurality of blades. A turbomachine and method are also disclosed.