Lower oil pan, oil pan baffle, upper oil pan, pickup, 1JZ engine mounting brackets if going in an 89+, BIC 1JZ engine mounting brackets if going in <89. If using a manual transmission, you would need a 1JZ bellhousing and flywheel from a 5-speed manual 1JZ with the R154 gearbox [1JZ manual or turbo Supra 5-speed manual]. You can also use the W58 gearbox [non-turbo Supra 5-speed] with a MkIV W58 bellhousing, clutch and flywheel, though this will not hold quite as much power as the R154. You can use the V160 6-speed gearbox from the TwinTurbo MkIV Supras, but you are going to need a manual transmission ECU, the gearbox and a custom driveshaft at least.
If going the automatic route, which may be easiest, the A340E 4-speed automatic from the Aristo/Supra may do you well. If your Supra that wants the transplant is using an auto tranny, this may be your best bet. There is a write-up on Arnout's page on how to combine the output shaft of the MkIII's A340 with the stronger A340E for the 2JZ. This will allow you to bolt the MkIII auto driveshaft right up to an upgraded A340E, and it will bolt right up to the 2JZ.
http://www.supras.nl/index.php?option=com_content&task=view&id=35&Itemid=52
1JZ electronics would not work. The 2JZ has many more sensors. You can do a 1JZ swap and put a 2JZ block under it for more displacement - this is the 1.5JZ and would be a 2JZ block with 1JZ head, turbos and electronics.