Solved! It's taken me half the morning to find this, posting the solution in case it helps somebody else out.

Find the "hosts" file (there's no extension) on the computer you want to access the sites - on Windows it's in C:\windows\system32\drivers\etc\hosts
on Linux it's etc/hosts

and add the following lines:

192.168.0.x www.mydomain1.com
192.168.0.x www.mydomain2.co.uk
192.168.0.x www.mydomain3.net

where 192.168.0.x is the internal IP address of your server.

You'll need to do this for every computer on your LAN that you want to view the sites.

See http://brisray.com/comp/cwserver2.htm for more information.