Hi Chris,
The simplest way to do this is to connect your PC and the other LAN device into a hub. Then you can use a program like wireshark or tcpdump to view all the traffic from the other LAN device. Another option is to use ARP poisoning, this would at least cause you to see traffic from other devices trying to leave the local subnet (by posing as your router).
If you have a managed switch that supports more advanced features you can use something like SPAN or port mirroring to get a copy of all traffic coming into one port, sent out another.
Cheers, James.