How do I wire three 3-way switches in one gang box to three switches in another box?
Basically you need three 14/3 cables between the two boxes. Wire each pair as a set of 3-way switches. 

Here is a basic diagram. Just do this for each set of 3-ways. The only difference is that you will need to pigtail three leads onto the incoming feed wire to feed each switch in the first box, and splice all the white wires together since they are all sharing the same circuit. At the box with the switch legs you will splice each white to it's corresponding white going to the lights.