Dryer connection changes from 10-3 with ground to 10-3 without ground
What a fiasco.  That was very bad work. 

Unless the ungrounded 10/3 is running inside steel conduit all the way back to the panel, in which case the steel conduit is the ground, and the work is complete.  However running cable inside conduit is the work of a dullard.  

The good news is it sounds like the ungrounded 10-3 is quite old work, making it grandfathered.  As such, you are allowed to retrofit a ground wire.  It must be 10 AWG, can follow any practical route, and must go back to either 


the panel the circuit is powered out of, 
a junction box served by another circuit served by that panel whih has a #10 or larger ground wire back to the panel, or 
somewhere along the Grounding Electrode System: the bare wires from the panel to the ground rods/water pipe.