I'm a four-trip WDW road warrior from Oklahoma, so I'll give you my two cents
First, I'll qualify this by offering that I
love the drive (and driving in general). So that's my bias. And that its gobs cheaper than flying, esp. for a crew of five.
We divvy up the drive into a longer first day, then a short second day. We head I-40 E to Memphis, then take I-22/US78 SE through Mississippi all the way to Birmingham, AL. We then loop around to I-20 E through Atlanta, then pick up I-75 south to Macon which is the end of our first day and roughly 14-15 hours time allowing for gas, food, and bathroom breaks. The second day is simply Macon into WDW, which is about 6 hours total. Don't forget to account for the time change (to EDT) once you hit Georgia (which is also why I make sure to get to Georgia the first day rather than the second - I spend that hour on the "long" drive day).
I've toyed several times routing around Atlanta through the "back roads," but I've simply never had the nerve to test any backwoods speed traps. I drive a bit conservatively by most standards, saying the limit + maybe 5 mph, and that keeps me from worrying about police.
Never charted it from Tulsa, but I believe you could head east to pick up I-40 along the Muskogee Turnpike, which I believe is a nice, divided, limited access highway most of the way. May not be four lanes both ways, but should be only moderately busy, and only about, what, 40 miles to I-40 (give or take)?
I-22/US78 is now
almost 100% interstate grade all the way to Birmingham, but the last stretch to connect into I-69 is not yet complete. Once I-22 ends, you simply take a service road over to the existing I-69 entrance, head south and loop around to merge with I-20E.
Exiting Memphis on US78 puts you on a somewhat industrial part of the road that I'm going to avoid this time by staying on I-40 all the way to I-69, then taking I-69 South into Miss, then cutting across one of several brief suburban stretches (about 3-4 mi) that link back into the "highway-quality" portion of I-22/US78.
The trip is great fun for us, and I hope you have a great time, too. Be glad to answer any other questions you might have.