does the board limit the current
It does but it will not limit it to a safe value either for the board or the LED.
s it really just a matter of providing each of my rows (or columns) with a series resistor on the pin?
No each LED requires it's own resistor, and the row needs a driver.
See:- http://www.thebox.myzen.co.uk/Hardware/Econo_Monome.html