1" or 1 1/2" PVC pipe, ~15' (probably best to get two 10' lengths in case you mess up)
4 PVC T joint pieces (or you can use an L at the top if you desire)
5 end caps
2 eye bolts (to fit on top of hinges, another kind of suitable bracket might work better if you can figure something out)
Construction is simple, cut pvc pipes into vertical pieces (about 13") and horizontal pieces (about 22"), four of each.
Put each long piece into the middle socket of a "T" joint and a short piece in one of the side sockets. Now connect one L to the next at a slight angle, about 5 degrees or so, in this fashion L L L L (see pictures).
Next, mark the location of the hinges on your rack and drill holes so you can screw in the eye bolts. You might have to make slight adjustments if you are too high or too low, or try using angles so that better contact is made. Also try filing the bolts down so they sit flatter on the hinges.
Finally, you can put end caps on all the exposed ends and glue all the joints using PVC cement, if you haven't already done so.
This rack is excellent for drying towels and takes up very little space, and all without drilling holes in the wall in case you happen to live in an apartment. If you want to make it a little more aesthetically pleasing, pvc is pretty easily painted with a brush or spray cans, so I think spraying the parts different colors before you connect them could look pretty cool. You would at least cover up the ugly writing that you can see in the picture.
Notes on the prototype:
I used 1" pvc for the whole thing, about 20' of it, and had some left over, but I think you need more than just one 10' length. My original design was based around using a series of Ls that fit loosely into each other and held together by gravity, but it wasn't nearly strong enough to hold towels and the Ls popped out of each other. Thus, one solid piece it was, with a slight angle between racks, so that towels can hang and not overlap too much. A swinging joint that is sturdy is ideal here, but I couldn't figure out a good way of doing it.
A suggestion would be to use thicker PVC, perhaps 1 1/2", on the upright parts, maybe even the whole thing, though it may take up too much space behind the door. This would probably help with the problem of the whole rack bowing when it is loaded with damp towels.
The second issue that could be remedied somehow is the attachment to hinges, which I managed with two 1/4" eye bolts. The roundness of the bolts presented a problem, they were constantly slipping off the hinges, even without towels, so I filed one side down to a flatter surface so it would lock on better. Getting the spacing between hinges right is key, so they both make solid contact and hold the rack up. Also, our prototype uses some poster tack/putty on top of the bolts to add a little more friction there. A better type of bracket could solve most of these problems, and would probably be flat and broad, so that it would really sit well on the hinge top.
--Edit 6-20-- Thanks to Seamus Dubh this problem is pretty much solved. Using a narrow screwdriver and hammer, I tapped the hinge pin upward about 1/4" or so to give the bolt something to hold on to, the rack stays on the hinges now!!
Good luck and leave feedback if you come up with ideas!