I have used Kaboom foaming bathroom cleaner on my fabric lampshades with outstanding results. I took the lampshades into the bathroom over the tub and sprayed them down good. I lightly rubbed the foam in with my hand and waited about 10 minutes for it to work. I rinsed very well and then took them out in the hot Florida sun to dry.
They all turned out bright and clean with no streaks. I am amazed how well this worked. I even used this on my lampshade with beads on the bottom, with great results. To think I was going to throw the shade out and replace it. Now it looks brand new.