I'm not sure that we can really claim that the country is more Republican than it used to be when four of the past six presidents have been Democrats, and one of the Republican presidents was barely elected by a minority of voters.
Perhaps the real effect is that the South has been able to increase its federal power as the population in states like Florida and Georgia has increased. This results in more representation in the House, which adds to political power in congress.
Immigration is probably going to continue to shape regional power in the future. Right now, Latino voters are gaining power, and Republicans are not very popular with that group at the moment.
Another regional shift has been the states that have better economies. Lately, it seems like the smaller, previously insignificant seeming states in the northwest have weathered the economic slump better than some of the formerly prosperous states. This can lead to population shifts as people come looking for work.
The major regional shift has been of people from the old "Rust Belt" in the Northeast and Upper Midwest to what is called the "Sun Belt" in the South. This has helped to make the country much more Republican since Sun Belt states are typically Republican.