Sounds exactly like you have a Vcc and GND connection swapped.
Still, I realized only when I press my finger around pin 19-21 of the first TLC5940, the 32 LEDs suddenly light up, just the way it should be.
22 is Gnd, 21 is VCC - could just be a poor connection to one or the other.19 is DPCRG, usually that is static, either hi or lo.20 is IREF, usually a resistor to Gnd.Check out your connections.
I'm gonna say "capacitors, add some"...
Alright, I gotta try tomorrow to see if that's the problem. So is needed to do is to put two ceramic capacitors onto the two IC chips based on this thread?