You mentioned that when you touch the BLANK pin that the LEDs all light up. What are you touching that pin with? If the BLANK pin is being held constantly high for some reason, none of your LEDs will light no matter what else is happening.
Have you tried resetting the Arduino by pressing S1 after the initial power-on. If you're using USB power to power everything, a voltage drop at power-on might be causing a lock-up state. A USB port is supposed to be able to supply up to 500mA, but you can never be sure. You could try measuring the current to the tlc5940's breadboard if you haven't already.
Also, just a note that one side of I(ref) resistor on the right side of the tlc5490 looks like it's very close to touching an adjacent pin.
Let us know what you find to give us more clues.