For starters you can visit my website for the 43rd Tennessee which was in Reynold's brigade at Vicksburg along with the 31st/39th, the 3rd, and 59th Tennessee, all East Tennessee regiments.
The Supplement to the Official Records, Records of Events, Vol. 67, pp. 272-285 contains some good information on their travels.
R. Donahue Bible of Mowhawk, Tenn. has written a book, "From Persia to Piedmont, 'Life and Death in Vaughn's Brigade." He specializes in the 31st/39th.
A regimental website is located at: http://www.tngenweb.org/civilwar/csainf/csa39m.html
This fellow may have information for you, though he could be researching the other 31st.
31st Tenn. Infantry Regiment email@example.com
If you can get to Vicksburg, the historian at the National Battlefield Park will have a few primary source documents including an abbreviated diary by Capt. George Hynds, Co. C.
Here's an interesting tidbit taken from my history of the 43rd:
04 15 1863 [Wednesday]
All quiet. Captain [James P. Burem] Beauram, [Co. G] 31st Tennessee Regiment and Miss [Netti M. ] Green of Mississippi were married about dark, by Rev. N. B. Goforth, Chaplain, 31st Tenn. The boys [43rd Tenn.] gave them a serenade. [Capt. Burem was killed at the Battle of Piedmont, Va., June 5, 1864, Confederate Veteran, Vol. 3, 1895, pp. 203, Netti filed claim for pay due Capt. Burem December 3, 1864, her lawyer later filed a query re this claim on February 17, 1865 at Spring Valley, Va., his body was never recovered, ". . . but his father erected a handsome monument to his memory in the family burying ground in his native state."] (M268 Roll 275)
The U.S. Army Military History Institute, Carlisle Barracks, PA lists the following in their unit bibliography:
laf Dec 90
39th Tennessee Infantry
Confederate Military History, Extended Edition. Vol. 10: Tennessee. Wilmington, NC: Broadfoot,1987. 806 p. E484C65.1987v10. (Contains numerous, scattered references to Tennessee units).
Crute, Joseph H., Jr. Units of the Confederate States Army. Midlothian, VA: Derwent Books,1987. Ref.See p. 304 (1 photocopied page) for a concise summary of the regiment's service.
Lindsley, John B., ed. The Military Annals of Tennessee: Confederate. Nashville, TN: Lindsley,l886. pp. 507-08 (1 photocopied page). E579.4L75. (Brief unit history).
Sifakis, Stewart. Compendium of the Confederate Armies: Tennessee. NY: Facts on File, 1992. pp. 148-49 (1 photocopied page). E579S53.1992. (Unit organizational history).
Tennessee. CW Centennial Comm. Tennesseans in the Civil War: A Military History of Union and Confederate Units.... Vol. 1. Nashville, TN: By the Comm, 1964. pp. 259-61 (3 photo-copied pages). E579.4A53v1. (Brief unit history).
This should be enough to get you going.