The X and Y axes will be moved by a captive nut on a piece of threaded rod. The nut is attached to a moving stage, and cannot turn. Therefore, when the motor turns, the nut is forced to move along the rod. To provide some flexibility in the system, the threaded rob is attached to the motor shaft by a tightly-fitting piece of rubber hose that I cut off my car.
These principles were all set out in the original McWire Cartesian Bot. I notice with chagrin construction of this is now considered a "death march"!
Some other design issues arose. The threaded rods in X and Y must be anchored in a thrust bearing. Now lego will never be able to survive having a threaded steel rod screwed into it for long. So the end bearings will be made of aluminium (because it's easy for me to work).
The actual rails and bearings for the XY stages to move on will be made of lego. They only have to support their own weight plus the object being printed.