Many of the principals of the war were dead by the time Edison's audio recording technology was operational, and many more had died by the time it was truly viable. While Edison did make recordings as early as 1877, these remained mostly experimental in the early years, with the recordings being done at his New Jersey laboratory. According to an 1886 item in Harper's Weekly, the early phonograph recordings were fragile and liable to become unusable, and only a few hundred phonograph machines had been produced before the mid-1880s. Even fewer had actually been sold.

By 1885, Abraham Lincoln, Andrew Johnson, Ulysses S. Grant, Winfield Scott, George McClellan, U.S. Secretary of War Edwin Stanton, Robert E. Lee, Confederate Vice-President Alexander Stephens, Nathan Bedford Forest, Stonewall Jackson, and all the Confederate Secretaries of War had all died. The most prominent living principals were Confederate president Jefferson Davis (died 1889), U.S. General William Tecumseh Sherman (died 1891), and Confederate General James Longstreet (died 1904). Given Davis's role in the conflict, he wasn't likely to be someone sought out to be immortalized, and I know of no effort to seek him out to be recorded. Likewise, Longstreet was not recorded, either. William Tecumseh Sherman, on the other hand, was recorded--but the recording does not exist. Or if it does, it is lost.

According to an article entitled "Testing the Phonograph" in the May 13, 1888, edition of the New York Times, Sherman was recorded during a phonograph demonstration by Thomas Edison the day before. The recording was made alongside a similar recording by Col. Robert Ingersoll of the 11th Regiment of the Illinois Volunteer Cavalry:

"There were strange sounds and sights in the unique headquarters of the Electric Club, in East Twenty-second-street, last night. Thomas A. Edison was there with a dozen of his perfected phonographs, and the compact little machines entertained a distinguished company of invited guests with oratorical eloquence and harmonious music. Gen. W. T. Sherman related war reminiscences, which were faithfully recorded and reproduced; Col. Robert G. Ingersoll said to a phonograph in the parlor: 'You are the most ingenious thing that was ever worked out of the human brain,' and the compliment came back with the clearness of a bell echo."

This was the last that's been heard of them, however, so unless by some miracle they turn up, the voice of Sherman (and Ingersoll) is likely lost forever.

Probably the most significant Civil War veteran to have his voice recorded was President Benjamin Harrison, who had served as a Captain of the 70th Regiment of the Indiana Infantry at the outset of the war, fought in Sherman's Atlanta Campaign, and participated in Sherman's March to the Sea. He was promoted to brevet brigadier general by the war's end. His voice was briefly caught on an 1889 recording by Edison.

President William McKinley was the only other Civil War veteran president to have his voice recorded. He served in the 23rd Ohio Infantry (incidentally, under the command of another president, Rutherford B. Hayes), though he was young and only ever served as a Private during the war. A couple different recordings were made of his voice, with the last being this 1897 recording.

On the Confederate side, as far as I know, the most significant veteran of the war to be recorded was Julius Franklin Howell. He actually wasn't all that significant during the war--he joined the 24th Virginia Cavalry in 1862 at the age of 16, and spent the whole war as a private--but he was significant after the war as the president of the Confederate veterans organization, the United Confederate Veterans. He lived to be 102, and gave a rather lengthy interview in the 1940s.

As far as descriptions go of the speaking style Civil War principals, there is quite a bit in the way of contemporaneous accounts of Abraham Lincoln's speaking style. A book that might interest you is Presidential Voices: Speaking Styles from George Washington to George W. Bush by Allan Metcalf, which quotes from contemporary sources of as many presidents the author could find. There's not a whole lot said about the actual voices of the other Civil War veteran presidents--more commentary about their oratory style--but Lincoln's description is fairly comprehensive. It starts with:

"...[Lincoln] attracted notice for his pronunciation, and there are some tantalizing tidbits in the notes of contemporaries. Horace White, reporting for the Chicago Press and Tribune on a Lincoln speech in September 1854, wrote that Lincoln “had the accent and pronunciation peculiar to his native state, Kentucky.” In early 1860, when Lincoln delivered his famous Cooper Union speech in New York City, a long way from home, another reporter noted, “He said ‘Mr. Cheerman,’ instead of ‘Mr. Chairman,’ and employed many other words with an old-fashioned pronunciation.”

"Lincoln was also reported to say git for get, thar for there, and heared for heard, and to speak with a “backwoods twang.” But early during Lincoln’s presidency, when young Robert Gould Shaw of Massachusetts visited him at the White House, Shaw heard a very different voice: “His voice is very pleasant; and though, to be sure, we were there only a few minutes, I didn’t hear anything like Western slang or twang in him. He gives you the impression, too, of being a gentleman.” Evidently Lincoln polished his language for use in the White House."

There's more descriptions that Metcalf pulls from, but he notes that they could often be contradictory, mostly depending on how the writer of the description viewed Lincoln politically. His supporters tended to describe his speaking style and accent favorably; his detractors unfavorably.

Metcalf has much less to say about Andrew Johnson or Ulysses S. Grant. For Johnson, Metcalf quotes from a Harper's Weekly description of his oratory while President:

"...When speaking before the Senate he appeared as a very different style of orator than when addressing the people. Before the latter his speeches are disconnected, full of repetitions, and not even his official position as Chief Magistrate of the United States is sufficient to keep him within the limits of good sense and decorum."

Of Grant, Metcalf simply states that Grant hated public speaking. A classmate from his boyhood remembered that he once had to make a presentation in front of the class and, "He spoke a selection from Washington’s Farewell Address, but he made fearful work of it."

Metcalf claims that the "remarks Grant had to make in public—as a general, secretary of war, and president—were unremarkable," and he was a much better writer than public orator. One peculiarity of Grant's speaking was that he made a conscious effort not to use foul language:

"He didn’t use off-color language, even in private. “I am not aware of ever having used a profane expletive in my life,” he once stated, “but I would have the charity to excuse those who may have done so, if they were in charge of a train of Mexican pack mules.”"

Above all, Metcalf writes, "Grant was known, if anything, for keeping silent."

As for William Tecumseh Sherman, while the Edison phonograph recording of his voice in his later years is lost, there are some descriptions of his speaking style in earlier years. One biographer noted that in his early years, Sherman emulated the speaking style of Thomas Corwin, Governor of Ohio and close personal friend of Sherman's father. Corwin was noted for his "Western oratory," of speaking fluidly, and being able to spin tall-tales. His oratory was marked by its "earthly wisdom and home-spun humor." The biographer says that Sherman "gained a reputation for speaking rapidly and endlessly, seeded with the same sort of humor and logic that had become Corwin's trademark."

On the Confederate side, a few brief descriptions of Jefferson Davis's speaking style were made. In one, he was said to have delivered a brief speech to some soldiers "in one of those off hand, chaste, eloquent, and appropriate fifteen minute speeches, for which he is so distinguished." In another, he was said to have spoken "with his usual vigor and point." In yet another, his was said to have spoken in a "full, round, sonorous" manner.

Robert E. Lee is a bit harder to pin down, because little was written about his speaking style during his lifetime, while after his death, the Lost Cause mythologized him to an extent that it's often difficult to separate fact from fiction. In one case, British Army Field Marshal Lord Wolseley wrote of an encounter of Lee decades after the fact, recalling he was impressed "with his old-fashioned manner of address." More contemporaneously, an 1872 item in Harper's New Monthly Magazine noted that Lee generally "wore a gravity and severity of manner toward all but his most intimate friends with the sallies which at odd times, however, he made in the happiest style". The same article would go on to describe Lee's speaking voice to be marked by "his peculiar slow, modulated tone".

EDIT: Brain fart. Swapped "principal" for "principle."