Not being familar with the Airport modem, or the myDlink site, all I can offer are some general suggestions:
Are your airport settings *identical* to the linksys settings? I am thinking about enabling PnP, making sure the ports are open on the airport, same security protocol? You may have to explicitly port forward the ports your camera is attached to.
From your description, it appears request to the camera through the myDlink router are being blocked, or perhaps the camera response is not getting past the router, or the dlink server.
You could try using wireshark to see if the camera are getting the request, and if they do, and they respond, then perhaps the router is not routing those packets.
As an alternative, you could bypass the mydlink server and use a service like dyndns.com to route requests directly to your isp/router to your camera.
Perhaps a posting to the Airport users forum might help?
J