The trees are Northern Arizona cedar trees about 1'-3' in diameter at trunk base, have a canopy of 15'-25' in diameter and I wanted to to put in trunk-surrounding raised beds that are 5' in diameter, 1' deep above ground. But when I do this normally (with no major tree involved), I dig rocks up out of the native soil, and essentially till it. What do I need to do or not do to keep these trees' roots damage-free ?
Most tree roots are within six inches of the surface where there is air and water that is freely available. Any kind of tilling or compacting will have a negative effect on the trees health.
Most mature trees like it just the way it is and changes to their environment must be done slowly, if at all, to give them time to adapt.
See your other question for better ways to do this.
You do not want to be adding a foot of soil to the base of a mature tree. That will suffocate the roots and likely weaken the tree. Maybe consider some alternatives. A bit of mulch and perhaps a few containers rather than an entire raised bed?