Heres a simple way that doesn't actually need the Arduino, you just need a one minute pulse to shift through the 60 shift registers, and the 59 output fed back to the 1 minutes, and the hours clock, but hey this is the Arduino forum 
each driver can drive a single LED, or a line of them.
Using the Arduino makes it easy to clear out the register at startup, and have manual fast forward to set the time.
The Arduino just has to shift one bit every minute ( and hour ) it doesnt need any lookup tables, or the shiftout library.
I havn't bothered with the notSCLR and the notG pins of the chips to save wiring, any dimming can be done with the LED supply, and clearing the register can be done with 60 zeros pulsed through at setup.