kevind2010:
My wiring will be greatly simplified if it's possible to run just one pin with a resistor in front of it for power, though my gut tells me that this probably won't work.
It won't work if you have only one resistor, but it will work with one 5V power supply (from the 5V power pin, not an Arduino output pin) and 16 separate resistors. If you think about it, with a single resistor, all the analog inputs will be connected together (draw out a circuit with, say, three phototransistors). As others have mentioned, you'll need an analog multiplexer to get 16 analog inputs into an Arduino.