So much depends on the underlying rock. In places where the roads are on sandy soils, they dry out pretty quickly. On roads with shales and siltstones underlying the roads, they can take quite a while, depending on temperatures and sunshine.
So maybe scout out potential sites with satellite before you head out. Red roads most likely are sandy and will dry faster. Gray roads not so much (and I'm not talking about the color of the lines on a map!).
Of course even if you wait until the roads with lots of bentonite dry out, you really have to keep an eye on the weather to avoid getting stuck. We were out mountain biking at Klondike Bluffs a couple of years ago and got hit with a surprise rainstorm while still on the trail (came up from behind the hill). Tough to even get back to the truck because of the clay around the tires. But even in 4x4, getting out of there was a tricky, slippery mess. Spent a lot of money at a car wash in town to get 2" to 3" of mud caked everywhere under the truck.
