Tripling the voltage will triple the current.
yes, but more importantly,
If you triple the voltage and that causes triple the current to flow then you get
9x the power!.
Feeding 9x the rated power into any device is going to end badly
EDIT2: Screw it, I've ran for 2 minutes using the 9V battery and nothing is over heating and my microcontroller is still rolling
What voltage is across the motor when its running? i suspect it's not 9V, the 9V battery probably can't handle the current and its voltage is dropping to something like 5V. If this is the case you might run into problems later if you connect a 9v power supply that can deliver more current.
The only other thing i can think of is that it may not be 3V motors, it might actually be rated for 3-6V or something like that.
I'm pretty sure i've seen motors that look the same as your one which are rated for 4.5V