I helped a fellow build a desk system on a new house once (a really long time ago), with a "simple" roofing system underneath the desk. I was the teenaged hired hand (stong back / weak mind) and the owner was some kind of nuclear engineer, turned home builder on the weekends. The deck was about 8 feet wide and ran with width of the house (maybe 25 feet). We constructed a lightweight roof made of styrofoam sheets (1.5 inches thick x 24" wide x 8' long), covered them with tar paper (I think he glued it) and reinforced it underneath with a 1" x 2" simple frame. These styrofoam roof "frames" were attached to the underside of the deck with hinges on the end closest to the house. On the outside end, the roof frames hung on detatchable wire hangers. The roof frames were sloped, which allowed the rain water to drain into a gutter system that drained away the water. As I recall the gutters were attached to the deck posts. The styrofoam frames could be lowerd periodically to clean off debris that fell thru the cracks in the deck. From the deck, you really couldn't see the roof system below (because it was black). This roof system was a little wacky, but it did work.
Basically, the idea of a lightweight roof that can be removed for cleaning does the trick!