I figured out what's causing the problem. The TX signal isn't making it to the ATMEGA8U2-MU that translates serial to USB. The TX pin on the ATMEGA2560-16AU is working, and signals from that pin make it all the way to the ATMEGA2560-16AU side of the RN4A resistor next to the ATMEGA8U2-MU, but the signal doesn't make it to the ATMEGA8U2-MU side of the resistor. It looks like the ATMEGA8U2-MU is messing up the signal.
Anyone else had problems with the ATMEGA8U2-MU?