Since someone else has mentioned CJs I will chime in.
I haven't taken my FJ40 on a trip yet, (I will be doing the four hour drive home in a week) but I can comment of driving this kind of vehicle on a long extended trip.
4 summers ago I took my stock 1979 CJ7 from our old home in Maryland up to Ottawa across to Lawrence KS down through New Mexico to our new home in Southern Arizona.
The trip lasted about 6 weeks, I had work along the way in Ottawa and Lawrence.
With their narrow track CJs are not fond of interstates that tend to be "rutted", so I mainly took old US Hwys. I traveled no more than 400 miles a day (except for the last day, 700+ because I wanted to get home). I camped out in state and national parks all the way. And anytime I saw another CJ I would stop and take a picture of them together.
Everywhere I stopped people talked to me about my old jeep.
It was a long trip. I know I smelled of gas and oil the whole time, and every night I felt like I had been road hard and put up wet.
I travel for a living and have since 1987, I have been to every state and all over the world. That CJ trip is one of my fondest memories.
I'd say if you have to time and the desire, you should do it.
I am planing an AZ to NC trip in my 40 this fall, I'm pretty excited about it.