Co. Aytch, obviously, which I have. Another I have is "Footprints of a Regiment" by a sergeant in the 1st Georgia regulars. I'm looking into getting the memoirs of Alexander Porter and Moxley Sorrel as well. Since there are no Union men on the list, if you have any feel free to share them. I'm wanting to build a collection of memoirs written by soldiers, and plan to eventually expand it to other wars.
Grant's memoirs are just fantastic -honest, candid, clear. Not really a common soldier's perspective, but valuable nonetheless.
My advisor will beat me if I don't mention the history of the 33rd Iowa that he edited. Unlike most regimental histories, it was written immediately after the war, and is thus a little less "cleaned up" - that is, more honest - than most works of its type.
One of the best for a Union Soldier is The Story of a Common Soldier of Army Life in the Civil War, 1861-1865 by Leander Stillwell. I've read it once but listened to the LibriVox recording of it 4 or 5 times. Both are free online.
A fairly high-profile Union example would by Joshua Lawrence Chamberlain's memoirs, which are now collected in three books:
The Civil War's a hobby field for me, and I have only The Passing of the Armies at present and can't speak to the other two. Passing addresses the end of the war, and most notably Chamberlain's famous order to his men to salute the defeated Confederate soldiers while they were formally surrendering their rifles. Modern readers tend to see the prose as a bit wordy and occasionally hard to follow, but no more so than you're probably encountering in other memoirs of the day, and possibly easier overall; Chamberlain was a good writer. He was also an outstanding autodidact and polymath who hadn't set foot on a battlefield before the war, but retired from the Army as a brevet major general and then went on to become governor of Maine and president of Bowdoin, teaching every subject in the curriculum with the exception of math and science.
Definitely one of the most fascinating people in the conflict, but in reading Fanny and Joshua, which is an excellent account of Chamberlain's marriage, you also start to get a sense of how living with someone like that can be pretty tough. (It probably didn't help that one of Chamberlain's war wounds -- a bullet through the hip at Petersburg -- may have rendered him impotent for the rest of his life, on top of being the thing that finally killed him 50 years later.) However, his wife, Frances, also influenced many of his political views and much of his direction in life. If you get interested in Chamberlain, then Fanny and Joshua and the collections of Chamberlain's letters would be worth your time as well.