Surprisingly little. The total casualty count is commonly accepted as around 3,000 but this number includes conflict with militia and military personnel in Georgia and pockets of resistance.

From what I remember the actual civilian casualties numbered around 100. The real damage was done to the city of Atlanta and the Georgian countryside where over 300 miles of railroad tracks were destroyed along with $100 million worth of property. Including a very significant amount of corn, cotton, hay, livestock (cattle and mules), horses and homes.

Considering the fact that Sherman managed that much destruction it's actually impressive how few civilians died, although they lost their homes and property (part of which if I recall correctly was due to some Georgian leaders not ordering evacuations) Sherman's march was remarkably disciplined and resulted in very few casualties considering the scale of the many other battles in the Civil War.