Drying Sunflower Seeds

Last year I tried growing some sunflowers in my garden.  They did quite well, and about halfway through the year I started thinking maybe I could get some seeds from them too.  They didn't look ready, so I waited patiently.  More patiently, it turns out, than the birds.  One day I showed up and almost every seed had been eaten.  This year I'm trying to pick them a little earlier and let them dry indoors.  Many sources recommended putting paper bags over the flowers after they turn downward, but that requires replacing the bags after each rain, and I don't have time for that.  Or a limitless supply of paper bags.  So here they are in my guest bedroom, probably picked too early.