I clean my wall hangings with a sticky roller. They'll also pick up dust. I'm going to go pick up three wall hangings from the quilter today that should never be washed. They have cotton, silk and polyester fabrics in them! The poly isn't like a cotton/poly blend either. It's from a bridesmaid dress so it's a fake satin kind of stuff.
I have one wall hanging that has a dimensional gathered lamp shade on it. No way for that one to be washed either.