Any tips on troubleshooting a chain of TLC5940's?

Follow the GND and VCC wires to the chip (on the schematic or the real circuit) and put the decoupling caps between those two wires near the chip. The rule is that any chip handling logic signals or loads switched from logic signals need decoupling capacitors. LED drivers handle high currents switched at high speed and need LOTS of decoupling. Ideally something like 220nF ceramic + 10uF ceramic/electrolytic next to each chip. Each cluster of chips (especially if remote from the power supply) should have something like 220uF - 1,000uF electrolytic as well to help keep the supply clean and reduce radio-frequency interference.

It can be an education to hold an AM radio near your digital circuit and scan the band!