Wow, girl, you rock! I've heard of trellising cantaloupes and hanging the fruits from the trellis with panty hose.
Artiom99, yes, I could see the vines stretched out over hay. Remember hay is messy! The worst part is putting it down. The hay will protect the melons and keep the vines and melons much cooler. Yes, you could let the vines grow on the bare concrete, but I think the vines and you! Would be much happier with something underneath them. But no plastic film. A light colored landscaping fabric would be suitable.