The computer running Squeeze Box Server and at least in my home network, has the storage drives connected to it, has an assigned IP address. The players may get a different addresses leased through DHCP, but not very often.
In the broadband router, there is usually a utility which allows you to associate a particular MAC address (the hardware specific address assigned at the factory) w/ an assigned IP address. Since DHCP, which assigns the addresses runs on the router, it will reassign the same IP address to a player over and over if you have it setup that way.
In Squeeze Box Server you can find the MAC address of the Logitech device.
I do agree that wired is less trouble and it also sounds better. As I think about it, there will be a MAC address for the LAN card and a MAC for the WLAN as well.
I detailed a method in another thread to keep the WLAN card from ever starting in the Touch.