Hello James,
The flashing WLAN light means activity (data getting transferred). Try disconnecting all your devices and check whether the light is still flashing or disconnect the router from internet and do the check.
Do you have your SSID as linksys or the default one? If your neighbor has the same SSID, then traffic could result from your neighbor's one. Try changing your SSID and password. Make your security WPA/WPA2.
Hope this helps,
Thanks
Vijay