Well, yes and no. 150 years is a long time, and we (Americans) have a different dialect, not to mention social norms, that influence the way we talk. I think it would be safe to say that Americans, in general, are more profane now than we were in the 1860s. This is the result of a softening of morals in terms of profanity, and the evolution of the art of swearing.

That said, people in the 1860s knew how to swear, and did so (especially soldiers). Fuck, shit, bastard, etc. were all in use during this period, and some were known to use these colorful terms more than others. We know that Gen. Joseph Hooker and Gen. Philip Sheridan on the Union side were known to be profane, and somewhat proud of it. How profane the soldiers were usually depended on their commanding officer, and that C.O.'s C.O., and the level of comfort people had speaking in this way around them. We know that soldiers on both sides tended to be formed up in regiments based on locale, so it also stands to reason that a New York regiment formed from city street toughs would be more rough around the tongue than one from a more rural community like one from Maine.

Primary evidence does support the fact that people from the 1860s were far less likely to put this profanity in writing, as it was definitely not seen as something a gentleman (or woman) would engage in. Journals, letters, etc. don't give us a whole lot of examples of direct profanity, although mention of it by others definitely comes up at times.

[Sources - James McPherson, 'Battle Cry of Freedom'; Bruce Catton, 'Grant Moves South']