It started for the most part in the 60s and 70s. Growing up in Tennessee during that time (and you have to remember that culturally traditional values are HUGE in that part of the country, what with it being the Bible Belt and all), as the Democrats became more and more wacko on social, cultural issues you did begin to see a marked shift in voting patterns during that time.
Oh, and don't forget the effect of Jimmy Carter's disasterous presidency.
When is the last time the Confederate states majority voted for a Democratic President?
Although the Republicans carried five ex-Confederate states in 1928, the Republicans didn't start making inroads into the region until the 1950's, when it won four ex-Confederate states in 1952 and five in 1956.
However, the South really began to turn Republican during the 1960's. In his book The Winning Side: The Case for Goldwater Republicanism (Putnam's Sons, 1963), Ralph de Toledano noted a strong Republican trend in the Deep South during the 1962 elections. The Republican vote was especially strong in the growing suburbs throughout the region.
Athough the Democrats managed to hold on to power that year by relying on an odd base consisting of blacks and rural segregationists, their strength was already seeping away. Ten years later, in 1972, Richard Nixon became the first Republican candidate to carry every Southern state, and since 1980, the South has been a solid component of hte Republican coalition.