Cheddy:
It is running on +5V and doesn't seem to mind. According to the board specs it will accept voltages from 3.3V to 5V, so I guess it has the voltage level shifter circuitry described above.It also runs fine without pull-up resistors on the I2C lines, which many say are a requirement for I2C comms according to what I have been reading. Maybe the Arduino Nano has built-in pull-ups.
So it has a voltage regulator for the power.
That doesn't mean it also has level shifters for the I2C lines.
The MPU-6050 might survive the (software only) pull up of the Nano.
Might be a different story if you connect another I2C device to the I2C bus, with hardware pull up.
Leo..