The official Mafia III Soundtrack has been released! It will be 101 songs, most of which are appropriate to the time period. There are a few that came out after 1968, some only a year or so after and others, like "Bottom of the River" by Delta Rae that date as far as 2014. You may remember that one from the E3 2016 trailer.
Others are remakes of songs from the period, such as The Avengers cover of The Rolling Stones song Paint It Black and Mourning Ritual's cover of Creedence Clearwater Revival's song Bad Moon Rising, both of which are also in the game.
The news broke today on the Mafia Game website. You can read their article Here.