Circuit design for controlling 6 individually selectable motors
Your initial thought is not bad: one H-bridge to determine the direction, and some kind of switch for each motor.
Forget triacs. They're only for AC, and besides will have a too high voltage drop at 6V power supply.
Relays would be the best thing, also because, thanks to the insulation between control and contacts, you can place them anywhere in a circuit. You'll just have to pay attention that this is not going to cost you more than a separate H-bridge for each motor.
MOSFETs are also often used to switch motors, but (especially power) MOSFETs have an internal diode which makes them unsuitable for bidirectional use, and besides, they're not symmetrical anyway.  

So the relays seem to be the remaining option, apart from a series of H-bridges. Like I said you'll have to check if they're the most cost-effective solution.