I want to move a kitchen range but moving the wire is a problem. The floor joists run the wrong way and I am planning on refacing the cabinets, so I can't access the walls. Is it legal to run a 220 line inside the cabinets for about 15 ft. If so, should I use conduit?

Running cable through the cabinets would be fine. Keep it up high, or out of the way, so that no damage can be done to the cable, say for example, from pots and pans rubbing or sitting on the cable.

Conduit would be a better job, esp to help protect the wiring, just need to have a junction box inside the cabinet where you change from cable to conduit. The conduit could end at the range outlet and eliminated any need for a junction box there.