There are lots of things you can use to tie an orchid onto a mount - jute string is my current favorite, as it seems to break down and disappear about the time the orchid has grabbed onto the mount. It looks natural and absorbs a little water, as well which might help the plant to not dry out too fast.
I also use bird netting cut into strips about 3in wide on the diagonal. It will hold a bit of spagnum or other medium around the roots if the orchid needs that, which is very handy. Sometimes I staple the netting in place, then tie the ends of the strip around the back of the mount, and they can be used for hanging, too.
Some people like florists wire, which has a plastic coating but still doesn't last too long in our hot, wet conditions. You don't want to use bare, thin wire as it will cut into the plant.
Strips of panty hose or knee highs, or any stretchy material is also a good choice and will usually break down in the outdoor conditions.
Some people use glue, something like Liquid Nails but I haven't had any luck with that. It sticks to the mount just fine, but the parts of the plant that absorb water just come unglued (so to speak).
