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.
The TPIC6B595 is the same as a 595 serial register, but has a built in LED/relay driver ( like a ULN2003 )
You can use any 3 pins to drive them, using Shiftout, and to set up the segment pattern I define a lookup table ( before the setup ) which looks like this for the pin connections I use from the TPIC to the segment :-
#define latchPin 11 // rck the pin you choose to go to the latch input of the TPIC
#define clockPin 10 // sck the pin you choose to go to the clock input of the TPIC
#define dataPin 13 // ser in the pin you choose to go to the serial input of the TPIC
const byte digitTable [10] = { // the patterns of segments for numbers 0 to 9
B11101110, B00101000, B10110110, B10111100, B01111000,
B11011100, B11011110, B10101000, B11111110, B11111100};
// and then at the end of the sketch when you want to display the data you just send something like :-
digitalWrite(latchPin, LOW);
delay ( 50 );
shiftOut(dataPin, clockPin, MSBFIRST, digitTable [ mintens ] ); // you can also have seconds if you like,
shiftOut(dataPin, clockPin, MSBFIRST, digitTable [ minunits ] );
shiftOut(dataPin, clockPin, MSBFIRST, digitTable [ hourtens ] );
shiftOut(dataPin, clockPin, MSBFIRST, digitTable [ hourunits ] );
digitalWrite(latchPin, HIGH);
delay ( 50 );
The TPIC has a latch, so you dont have to refresh the display ( and you can take a video of the display without the scrolling bands moving down )