I thought I'd share some pictures of my solution, inspired by Hal's great instructions. I was shopping for the soap-dispenser when I found this basket and realized that it would work pretty well for what I wanted.
Like others have mentioned, this doesn't give me a hands-free setup but I don't mind because I prefer to turn the water flow down to trickle and it would be harder to do that if I had to turn around each time.
The basket I used is at Bed, Bath & Beyond, link here:
http://www.bedbathandbeyond.com/prod...p?SKU=17859200
I found the condiment bottles at Wal-Mart - I expect they could be found anywhere. They happened to fit perfectly in this basket.