I'm going to limit myself here to talking primarily about the Union, since I can't speak to the Confederate situation. To the extent that I address the South, it will be within the context of southerners attempting to obtain compensation from the Federal (i.e., Union) government.

The Civil War was indisputably the most destructive war in American domestic history. It occasioned not only a massive loss of life (by the standards of the American nineteenth century, the death toll was apocalyptic, and its social and cultural consequences have been well documented by historians like Drew Gilpin Faust in This Republic of Suffering and Michael C.C. Adams in Living Hell: The Dark Side of the Civil War), but also a horrific economic cost, damaging or destroying the property of thousands of - more or less innocent - civilians who were caught in the cross-fire. Wherever Civil War armies existed, they destroyed property. In camp, on the march, or in the field, they stole food, trampled crops, chopped down forests, and burned and looted homes. In the South, where the majority of the campaigning occurred, the majority of the destruction occurred also. But civilians in Union regions like western Maryland, south-central Pennsylvania, West Virginia, Kentucky, Missouri, and Kansas suffered, too.

Nineteenth-century America was a profoundly litigious society, and Americans responded to this crisis in much the same way as they responded to other crises: They called their lawyers and petitioned their representatives. Making claims for damages - whether to military authorities, courts, or administrative bodies - required time and paperwork, including witness testimony, affidavits, and property valuations. Whether and how property owners were able to obtain compensation for property destroyed during the war depended on who and where they were, and how their property came to be destroyed.

Through the congressional War Claims Committee and the Court of Claims (predecessor to the modern Court of Federal Claims), northerners could petition for restitution if their property was damaged or appropriated by Federal troops. Claimants filed for restitution of property taken by Federal troops or destroyed in battle, crops and livestock killed, and even land damaged by mass burials or encampments. But these claimants faced obstacles. The burden of proof was extremely high, and there were certain kinds of damage for which the Federal government simply wouldn't reimburse. Samuel Mumma's farm near Sharpsburg, Maryland was the site of some of the most heinous bloodshed during the Battle of Antietam in September 1862. His home was burned to the ground by Confederate troops in an attempt to prevent it from being used by Union troops as a nest for sharpshooters. Mumma's post-war claim for damages was denied on the grounds that the damage had been done not by Federal troops, but by Confederates. William Roulette, another Sharpsburg-area resident whose property was caught in the middle of the fighting, was similarly unlucky with thousands of dollars' worth of damage claims he filed.

Of course, the Federal government was not the only game in town. The Commonwealth of Pennsylvania, for example, accepted claims for damage done by Confederate forces. An index of the hundreds of claims made by the residents of York County has been digitized here. But it's not clear how successful the claimants were at obtaining payment. Local historian Scott Mingus claims that they weren't successful at all. Sometimes, reimbursement was provided through one-off legislative appropriations, as when the government of Pennsylvania appropriated several hundreds of thousands of dollars in a series of discrete aid packages to the citizens of Chambersburg, who were the victims of a particularly destructive Confederate cavalry raid in July 1864.

If you were a southerner, unfortunately, you were probably a helpless case. An 1864 act of Congress limited the jurisdiction of the Court of Claims to northern states. The Federal government refused to compensate property owners who were disloyal at any point during the war (residence in a southern state was considered prima facie evidence of disloyalty, and the burden was on the claimant to prove otherwise), and would not even compensate southern blacks and loyal southern whites until the establishment of the Southern Claims Commission in 1871. The Southern Claims Commission itself compensated citizens only for property appropriated by the military, not for war damage. Approximately two thirds of the 22,000 claims made to the Commission were denied. It eventually paid out a total of about $4.6 million before wrapping up its work in 1880.

EDIT: Corrected a couple of typos.