In a word: nervous. Excluding Grant, who conceived of this idea of a Union army cutting loose from its supply lines to live off Confederate spoils, most were unsure if it was even possible.

In a humorous bit of irony, although Sherman is famous for the tactics involved in his "March to the Sea," Sherman had originally been the strategy's biggest doubter! During the Vicksburg campaign, Grant had tried about half a dozen different strategies to take the Confederate city/stronghold (these involved trying to re-dig the Mississippi River, going down the Yazoo, and direct assaults). He only found success when he marched south, crossed the river well below Vicksburg, then marched east to Jackson, and back west to Vicksburg. To pull this off Grant had to get through the swampy marshes below his base of operations, and to accomplish this, he had to abandon the majority of his supply train. Sherman thought it was a nutty idea (unhitching from your supply lines to march blindly through enemy territory was practically unheard of in the conflict), but Grant thought the Confederate farms and fields on the eastern side of the Mississippi would be able to sustain him...and he was correct.

With this blueprint, Sherman eagerly attempted to replicate the feat on a much larger scale. Of course, Grant had every confidence that Sherman would be able to barrel through Georgia, but Lincoln and the administration were only convinced after Grant explained that if anyone could pull it off, it was Sherman. Lincoln gave the final OK, and Sherman set off. Believing that the less the papers or spies knew about what he was actually up to, the better, Sherman broke off contact and rarely wired in updates. This meant that the anxiety over what was going on with the army under his command only grew in Lincoln's cabinet. There was a lot of nervous grumbling about this "lost" army following Atlanta's fall, as they did seem to kind of disappear in the eyes of those hundreds of miles away (again, few telegraph dispatches came through, and Sherman, always distrustful of reporters, made sure nobody was reporting about his movements if he could help it).

So, long story short: there was a lot of apprehension. Once Sherman reappeared in Savannah having accomplished the feat, Lincoln and his cabinet were ecstatic. It was a clear indicator that their armies were in dominant control of the western theater, and a crushing blow to Confederate morale and logistics.

[Sources: Doris Kearns Goodwin, 'Team of Rivals'; James McPherson, 'Battle Cry of Freedom'; Bruce Catton, 'Grant Moves South']