Semiconductor device packages formed in accordance with methods of packaging semiconductor dice in grid array-type semiconductor device packages using conventional lead frame or lead lock tape assembly equipment are disclosed. Circuitry-bearing segments having an electrically insulating layer that carries redistribution circuitry and redistributed bond pads and which is supported from beneath by a support layer are secured to the active surface of a semiconductor die. The support layer may comprise an electrically conductive material, which may act as a heat sink or as a ground plane for the packaged semiconductor device. The methods provide increased accuracy with which segments are placed on a semiconductor die relative to the placement accuracies provided when pick-and-place equipment is used to position conventional grid array substrates relative to semiconductor dice.