Was there a call for such trials from union political leaders or the populace? It seems that the confederate leaders got off fairly easily for conducting an armed rebellion and engaging in what could be considered war crimes. Was it felt that holding confederate leaders to account would be counter-productive, turning them into martyrs for the southern cause?
In was Lincoln's expressed wish that such tribunals not be held. During the River Queen conference that Lincoln held in March 1865 (this was on the steamer 'River Queen'), he told Gen. Grant, Gen. Sherman, and Admr. Porter his intentions for handling the peace negotiations. Lincoln wanted to start the work of reconciliation and healing as soon as possible, and told his three commanders that all he really wanted was for the Confederates to lay down their arms, pledge allegiance to the Union, then be allowed to go home. He didn't want to prosecute the generals, and except for Jefferson Davis (who Lincoln hoped would escape to Mexico or another country), he seemed content to let the civic leadership slide. Lincoln thought that the quickest way towards a lasting peace was via an olive branch, and as soon as the states in rebellion agreed to return to the fold (and to do so acknowledging and respecting the 13th amendment), he wanted to welcome them. It's hard to know what he had intended for Reconstruction, or how he might have handled the troubles that erupted over voting rights, etc., but we do know from this River Queen conference that Lincoln absolutely did not want tribunals.
Grant and Sherman agreed, and both wrote later that they admired and respected Lincoln's decision. This makes sense, too, for as rough as Reconstruction and the ensuing years were as it concerns north vs. south tensions relating to the war, a harsh peace full of military trials and hangings would have likely been a disaster (the divisions in the country likely would have been much worse). Of course, this veers off into history what-if territory, but to answer OP's question: there were no trials because Lincoln said so (and the military leadership agreed). The Radical Republicans might have felt differently, but when Pres. Johnson took over, that ship sailed.
[Sources: Doris Kearns Goodwin, 'Team of Rivals'; Bruce Catton, 'A Stillness at Appomattox'; Jay Winik, 'April, 1865' U.S. Grant, 'Memoirs']
Tangential: Confederate general Robert E. Lee was stripped of his citizenship after the war. Citizenship was restored to confederates in a mass-pardon in October 1865 - except Robert E. Lee, who was required to apply separately. Though he did apply for citizenship, it was not restored until 1975 under Gerald Ford.
http://www.politico.com/story/2010/07/house-restores-citizenship-to-robert-e-lee-july-22-1975-040085