I used to use a dlink router and had the exact same issue you are having. I bought a Linksys router and never had an issue afterwards.
You need to go into your router and see if there is an option to turn off the NAT. If not, there may be a user name and password you can use to sign into your router that will allow you to change more advanced settings. Usually your ISP are the only ones that can provide you the information though. If you google search your model router for the ISPs user name and password you may find it though.