A method and apparatus nondestructively remove a cut segment of predetermined shape from a limp material sheet workpiece and transport the cut workpiece segment to a staging area for further processing. Such removal and transporting is accomplished without distorting the shape of the cut segment. An apparatus (10) includes a cutting table (12), a dispensing and cutting gantry (14), a picker gantry (16), and a sheet material dispenser assembly (18), all under the control of a controller (20). Picker gantry (16) includes a cover segment picking mechanism (16A) that removes a cover sheet segment off a workpiece material sheet. The picker gantry further includes a workpiece segment picking mechanism (16B) for removing a workpiece segment from the workpiece material sheet.