Hi omni96,
Thanks for your response. I'm checking this a bit late (forgot to press the Notify button :*).
Anyway: I figured out why it didn't work. I started a Windows command-prompt and with 'mode COM4' I got more info on the serial port. I noticed that, when I started Serial Monitor, the values of DTR and RTS changed!
I don't know why, but it seems like the Uno doesn't bother and the Leonardo does...
DTR must be set to ON and RTS must be OFF. I changed the code in my Serial class and voila: it works fine 
Hope this helps you too!