How to use arduino for some semi-complex switching?

Hey guys,

I'm working on a project for my vehicle's amateur radio setup that would allow me to use one mic for each of my radios. Of course, the mic would only transmit to one radio at a time.

The Mic has 4 wires (PTT (push to talk), Mic +, Speaker +, and ground).

I know I could use a 4 pole 3 position rotary switch for this but I'd love to use arduino so I can display which radio I'm routing to on an LCD. I have several rotary encoders on hand that I feel would be nice to use to cycle through the radios.

I'm just having a hard time wrapping my mind around how I can do this elegantly. I've considered using the arduino to switch on and off several relays but that would require branching off each of the mic's wires to several different relays. It just seems kind of sloppy to me.

Any help would be great! Thanks.