As the MakrBBot III only has a LCD display and controller, there is no easy way to manually level the bed. I used just to connect it to my computer, and use Pronterface to manually move the nozzle to all four corners and adjust the screws. But  this does get annoying, and it would be nice just to do this from the controller. Apparently Marlin 2.0 has a menu option to move to the corners, but I just reverted to Marlin 1.19.

Then I found this video from Martin Zeman where he comes up with a very simple solution to this. It is a simple gcode file that just moves the nozzle to all four corners, pausing in between movements so you can adjust the bed.

; Bed leveling with absolute positions for MakerBot III
G90 ; Absolute Positioniong
G28 ; Home all axis

G1 Z5 ; Lift Z axis 5 mm
G1 X15 Y15 ; Move to first corner
M0 ; Pause print to place paper under nozzle
G1 Z0 ; Z axis to 0
M0 ; Pause print

G1 Z5 ; Lift Z axis 5 mm
G1 X15 Y135 ; Move to second corner
M0
G1 Z0 ; Z axis to 0
M0 ; Pause print

G1 Z5 ; Lift Z axis 5 mm
G1 X160 Y135 ; Move to third corner
M0
G1 Z0 ; Z axis to 0
M0 ; Pause print

G1 Z5 ; Lift Z axis 5 mm
G1 X160 Y15 ; Move to fourth corner
M0
G1 Z0 ; Z axis to 0
M0 ; Pause print

G28 ; Home
G1 Z10 ; Lift Z axis 10 mm

It's the same as in the video, only I added a pause after the move and before lowering the nozzle so you have time to put the piece of paper in place. 

You probably would have to heat the nozzle first so there is no hardened filament at the nozzle tip. I suppose you could add this to the gcode as well, but I always pre-heat manually through the menu.