I would use 6 TPIC6B595 latch/drivers...
I was looking at those, but there were two problems I thought I may have with them.
First of all, I can't seem to find any decent tutorials for these 595s. They are fairly new to me and I would have to figure out how to convert a numerical value to a specific pattern of I/O to drive each of the relays. I also already have the code figured out for my original idea that didn't require these 595s.
Another issue I just thought of was that I'm using a ChronoDot for the time keeping, and it uses those same clock and latch pins that these 595s would otherwise seem to use and I'm not sure if they would like to share with the ChronoDot.