An interesting question and a great answer from Stephanie.
I would imagine the wonderful people North and South Carolina would be upset that they have been left our of your definition of the deep south. My experience has been that southerns are a proud bunch and each of them thinks their state is the best. Even though I was born and grew up in Ohio, my mother was from South Carolina and I spent every summer of my childhood there.
Chris, I hope that you and Nikki will make this trip-- How much time are you planning? That may help craft my answer.
I know you have been to the United States and know that each state has its own personality.
I agree with Stephanie that Louisiana is in a different class than the others. Louisiana is very diverse and rich in its history and culture. Yes, it is in the south and has many of the southern US characteristics but because of the Spanish, French and Caribbean influences from the past you have a melting pot of cultures. Most people know of Louisiana because of New Orleans but what they fail to understand is that the entire state is beautiful and has a varied topography. Louisiana offer rives, lakes, bayous, marshes and rolling hills.
The focus of your trip will more than likely be the fantastic city of New Orleans. It is alive with music and food. Some of the best chefs in our country are located in New Orleans. The French Quarter is a few blocks in each direction but there is a lot going on in those few blocks. Moving around the city and seeing the sites is easy.
Tennessee is spread out and from my perspective each city has its own personality. Nashville, Knoxville and Memphis are the biggest cities. Nashville is the home of country music. You can tour Graceland (the home of Elvis Presley) go the the Grand Old Oprey and the country music Hall of Fame. Knoxville, is my least favorite, but very close to the mountain TOURIST town of Gatlinburgh. Gatlinburgh can be fun for a couple of days but it is crowded and touristy. Dolly Parton has an amusement park nearby in Sieverville. Lots and lots of outlet shopping in this area. It is a shoppers Mecca. Finally, Memphis has a real southern feel and the famous Beale Street where you'll find wonderful food, bars and music. Memphis is the home to the blues. Lots of great music.
Alabama has some beautiful beaches. Montgomery and Birmingham are the two major cities. They are nice towns but if I were going to Alabama I'd spend my time at the beaches. Green rolling hills and lots of good BBQ.
Georgia as Stephanie pointed out offers the beautiful town of Savannah. Definitely worth a trip. The architecture is lovely and the people are hospitable. You'll find a wide array of southern foods-- rich sauces, fried foods and maybe some grits, biscuits and gravy. If you are considering Savannah, and you should --- please watch the movie, "Midnight in the Garden of Good and Evil" It will give you a taste of what Savannah has to offer. You could spend a few days in Atlanta enjoying the city life and a few days in Savannah. They have a nice jazz festival each year. There are some nice beaches nearby.
The one place to consider that was not part of your original question is Charleston, South Carolina. A beautiful southern city not to be overlooked. I can recommend a B&B that will provide everything you need. This town oozes southern charm. Lots of nice restaurants and the beaches are nearby. You can have a few days of history and then rent a beach front home or condo.
What month are you planning to do this? They may influence which location you select.
Enough for now.
Let us know where you decide to go.