Here is what I did:
(1) I went to Home Depot and found a 'poster board' size piece of 1/2" plywood and had them trim the piece down to "14x11". (The wood cost about $5, and they will cut it to size for free!) Next, I used our left-over white trim paint to give the wood a shabby chic look. I was nervous it was just going to look a whole lot of shabby and not so much chic, but I think it turned out nice.
(2) After the paint dried, I positioned my vase (an empty pickle jar) on the board to figure out where I wanted it to hang. When I got it centered, I drilled two holes on each side of the jar.
(3) Next, I wrapped hanging wire around the top of the jar and laced the wire through the holes. I tied the ends in a knot and clipped the excess wire.