The Anthro wine rack version I saw consists of a long jute bag which is narrower than a wine bottle. My woven cotton table runner is wider than a bottle but you could always trim yours to be narrower. Since I don't own a sewing-machine or overlocker, I just went with what I had. The dappled fabric is very forgiving but the stripes have both pro's and con's. The pro being that the stripes act as a guide to help you to sew straight - I didn't even use pins. The con is that it accentuates the shape of curvy bottles and if you don't sew straight it really shows when you put the bottles in. However, if you have but a smidgen of seamstress in you (or just own a machine!) this should not be a problem at all. Also, the firmer the fabric the straighter the final product will hang. Choose thread that blends in with the fabric and that will camoflage any imperfections.
You will also need a strong pipe. I'm not sure whether a wooden dowel or PVC pipe would carry the weight, so I used a metal pipe that I spray painted. An old broomstick might also work.