Can I swap a USB power sources for a 3-AAA battery source?
My question: Is anything bad going to happen if I hook up 5V USB
  power-pack in place of the (3.6V total) batteries?


The LEDs will likely fail


  If I need a resister in series, how do I tell what resistance I need?


Measure the voltage when operating on new batteries and note it down. Swap to the 5V supply but put 1 kohm in series and gradually lower that 1 kohm until you see the same voltage as measured previously. Use that modified value of resistor for each LED lamp you have i.e. they will all require their own individual dropper resistor.