Need help with eliminating LED-generated noise
The culprit is not the LEDs themselves, they're harmless, but the TLC5940s, which switch at high frequency to control LED brightness through PWM. You can't filter the PWM outputs (you can, but then the brightness control doesn't work properly anymore), so that's out, but you can try to do something about decoupling power supplies. Not guaranteed to work, the fact that the scope's probe picks up the signal unconnected indicates that it's probably radiated, but it's worth trying.
Decouple the TLC5940s properly. They have to provide a lot of power, so that means 100\$\mu\$F, 1\$\mu\$F and 100nF all parallel on the power supply for each device, the smallest value closest to the pins.
Decouple your 5V power to the microcontroller properly: 100nF close to the pins.