So the noise has nothing to do with the DAC chip. I put 2 10k resistors in a divider arrangement, giving 2.5v, the same as the output of the DAC when no notes are playing. Then connected the output jack to the divider, bypassing the DAC, and I still get the noise with MIDI message reception.
I'm stuck... :~
Tomorrow I'll try making my own MIDI input circuit on a breadboard, to see if the sparkfun board's an issue.