when I connect them to the 5V and GND pins on the Arduino, the LEDs turn off
As pillard said yes, for an explanation see:_
http://www.thebox.myzen.co.uk/Tutorial/Power_Supplies.html
Also see:-
http://www.thebox.myzen.co.uk/Workshop/Motors_1.html
and
http://www.thebox.myzen.co.uk/Workshop/Motors_2.html
couldn't a transistor eliminate the need for diodes
No, you need a diode.