There's a good chance that the network address changed when you changed provider equipment and this may be the cause of your problem.
If you have the CD that came with the DNS-323 there is a setup utility on that, if you don't have the CD, it can be downloaded from D-Link support - install that and see if it finds the NAS, if it does it will let you change the network address.
Another possibility is that the old provider equipment allowed you to use a gigabit Ethernet connection and the new equipment doesn't - if you had manually set the speed to gigabit (and not auto), this could prevent you from connecting if the new equipment does not support gigabit speeds.
If the setup utility doesn't work, look at the back panel of the NAS - there is a pin hole reset switch alongside the network port - use that to reset it to default and set up again.