When DH first proposed this idea to me before our March 2011 trip, I was pretty against it, but agreed to try it. Our trip is 1400 miles, so pretty much 24 hours door to door. We left at around 5 (after school) on Thursday evening, stopped for dinner an hour or so later, then stopped again to get the kids into their PJs. DH drove all night listening to various podcasts while I snoozed, and then I took over after breakfast in the morning and drove most of the rest of the way.
It actually worked like a charm, and now I prefer it to the hurry hurry hurry WAIT stress of flying. We have a Honda Odyssey so there is plenty of room for us and our 2 kids to bring lots of things to entertain us. The DVD player is great to entertain them, as well as books, coloring, the iPad and my daughter's iPod Touch. We also stop a lot, I doubt we go for longer than about 2 hours during the day without stopping to stretch our legs, get a snack, go to the bathroom, and change DS's diaper.
Overnight DH was able to make great time. Since the rest of us were sleeping, he didn't have to stop so often and was able to keep up a good speed. He is able to go without sleep pretty easily, and didn't have any trouble staying up with the help of Monster energy drinks and the podcasts. I did make him promise (even though it's obvious) that if he was tired or worried we'd pull over and stay somewhere.
The first time we did it last year we just stayed at a cheap hotel on I-drive for the first night, and then continued to Disney the next day. We got there at 6 Orlando time (after the time change) and were all probably asleep by 7. Then we slept in the next day until around 8 and took our time getting over, checking in at BWV, and heading into Epcot leisurely. After that recovery time we didn't have any fatigue issues, so I'd recommend giving yourself some time to recover from the overnight without a lot of rope drop stress on yourselves!
This year we're doing it again, although we are going straight to BLT when we get there. Then we'll sleep in and get to the parks on when we get there!