Grant was the one who influenced Sherman to depredate/live off of the civilians. Sherman was aghast at how his troops would steal from civilians, early in the war. He was always at a loss on how to stop his men from plundering, till Grant showed him that to campaign, living off the land, and get away from supply depot's, to win.
Grant was calling for continuing forward to Vicksburg, after crossing the Mississippi below Vicksburg with his army, Sherman is worried about supply lines, and notified Grant of his concerns, Grant answered back...
"I do not calculate upon the possibility of supplying the army with full rations from Grand gulf. I know it will be impossible without constructing additional roads. What I do expect is to get up what rations of hard bread, coffee, and salt we can, and make the country furnish the balance."
Afterwards, when building a NEW bridge over the Big Black River, Sherman went to sleep, waken at the news that Grant had arrived, Sherman greeted his Commander and said:
"General Grant, I want to congratulate you on the success of your plan. And it's YOUR plan, too, by heaven, and no body else's. Nobody else believed in it!"
It wasn't until Grant showed this example proven by flanking the Vicksburg garrison, and hitting them from the east, cutting off their communications, that Sherman finally accepted the idea...and then he adopted it whole heartedly from then on.