hollyjester:
I did not use diodes because my teacher gave us a budget. However, due to the low voltages,
Without the diodes the voltages will be high voltages - inductances/motors generate as high a voltage as they need to to pass current through the external circuit - without the diodes to provide a safe current path those voltages may damage the transistors. Even 1N4148 diodes would probably do as they have a pulsed current rating of 450mA, and they are cheap.
Another issue with the H-bridge is that it has shoot-through (both NPN and PNP are on simultaneously (briefly) during the switch-over, directly shorting the supply rails together). If you use 4 pins to control the 4 transistors you can introduce dead-time to prevent this.
If not using PWM then it won't be wasting much power though. Bipolar transistors carrying large currents can take several microseconds to turn off, much slower than the turn-on, due to minority carrier storage effects.
However the issue of the transistors getting hot is most likely because there's not enough base current to drive them into saturation (current gain drops off rapidly at higher collector currents, and saturation requires more current still) Try base resistors of 220 ohms.