Yes it's OK to plug a 36V battery into a MP III and it will not harm the battery or controller, but you may have a problem if the voltage setting on the controller has been set to 48V as this will cause the controller to cut out very quickly (or not work at all) due to the lower voltage of the 36V battery.
I seem to recall that the original MPIIIs were supplied with their controllers set to 36V by default, so hopefully you should not have a problem.
I have all my controllers set to 24V so that I can use any pack between 24V and 48V without having to reprogram the controller every time I use different voltage packs.
Alan