A method is provided for disassembling a rotor of a gas turbine engine. During this method, the rotor is provided which includes a rotor disk and a plurality of rotor blades arranged circumferentially about an axis. The rotor blades include a plurality of airfoils and a plurality of attachments that mount the rotor blades to the rotor disk. Each of the rotor blades includes a respective one of the airfoils and a respective one of the attachments. A press is arranged against the rotor. The press axially engages each of the rotor blades. The press moves axially along the axis to simultaneously push the rotor blades and remove the attachments from a plurality of slots in the rotor disk.