On a few occasions I've seen the Yamaha's get upset about something, and removing power from the Yamaha for a few seconds brings it back online.

Has the system ever worked for you? If you changed the router, the new router's settings may not be appropriate for the configuration expected by the RTI system. You'll need to guess what the former router's settings might have been.

My approach (if I blindly walked into an installation such as this) would be to use FING to see what is on the network, assume that everything associated with the RTI system was set for fixed IP addresses and define the router's DHCP range appropriately. Regardless, some devices, such as Sonos must be able to use the DHCP server.