I have the same issue and grey as well. I drive with the windows down all the time (including the rear) because of my dogs.
I found that a good shop vac w/a soft bristle brush attachment removes 99% of it. Try that first BEFORE you get anything wet. You may find the dirt/grime will just will adhere itself to the material if you do. The headliner material is not very hardy and a bit cheap.
When I used my shop vac on it for the first time, I could make a racing stripe from one end of the truck to the other.. kinda gross. I have to clean the headliner about every couple of months & found that leaving the rear open all the time causes most of the dirt coming in. Closing the rear window at higher speeds helps a bunch.