I had a neighbor who bought a Missouri Fox Trotter and the former owners were breeding her to get mules. So I had always assumed the gait came from the mare side of the equation.
I also own a 1/2 Fox Trotter yearling. Daddy is a QH but Mom is my Fox Trotter mare and the man responsible for this union said that the foals usually take the gait from the mares. (He bred several Fox Trotters to his QH stallion). BUT, my foal only seems to walk-trot-canter, so I don't know if he will gait, probably not. But the guy who bred my mare says her 3 yr old foal by the same QH stallion is "pacey." So I personally don't know, but I thought sometimes the gait was passed on even if one parent wasn't gaited.
There are also "crop up" gaited horses in normally non-gaited breeds such as Morgans, Appaloosas and Saddlebreds I believe. So that probably would not be happening if both parents had to be gaited?
But honestly, I don't know. This is just a topic for discussion that I am interested in learning about too.
To the original poster- I like to think of gaited horses as horses with extra "gears." Like most horses walk-trot-canter. Gaited horses often walk-trot-canter-pace, and have an intermediate, smooth gait as well, which can vary by breed. For instance, my Missouri Fox Trotter walks, flat walks, fox trots, paces, running walks, and canters. It's like I have all these extra gears that I don't always know what to do with.
It is genetically reproduced but it can take some training to get a horse "set," or reliable in it's intermediate, smooth gait. In other words, the horse is born with the ability, but it may take some practice to get the horse doing it steadily on command. That's the problem with my girl. She has all these gaits and I have trouble holding her in the gait I want. But it's no biggie because we just have a blast trail riding and I really don't care. I enjoy her just the way she is.