Here is a proper method to use your default gateway (router DNS) as the primary DNS configuration. However, if your router has a misconfigured DNS, this may fail as well. This is an unlikely scenario if your router automatically retrieves DNS settings from your ISP.
Start -> Search
View Network Connections
Double-click Local Area Connection (Realtek PCIe GBE Family Controller)
Double-click Properties
Double-click Internet Protocol Version 4 (IPv4)
Select the checkbox "Use the following DNS Server Addresses"
Enter the following values:
For Preferred DNS Server: 192.168.1.1
For Alternate DNS Server: 8.8.8.8
Click "OK".
If this does not work, try using 8.8.8.8 as the primary DNS and 192.168.1.1 as the secondary. This may prevent you from connecting to other computers on your home network, as it will use Google's public DNS servers. It can, however, restore Internet access.
Getting Additional Support/Information from your ISP and Cisco/Linksys
If you continue to have problems, you will need to access the router itself and check all configuration settings. This can require some intermediate to advanced knowledge of networking protocols, and you may want to contact either your ISP or Cisco/Linksys for further assistance. Your ISP may not support 3rd party devices like routers, and support at major ISPs usually say this so that they can cut costs. You may still want to try them first, telling them that you need to reconfigure your router and need configuration information. You can then take down this information, and consult the manual for your router or contact Cisco/Linksys support for a walkthrough on setting up the router.
Those of us who have configured these routers do not have much difficulty doing so, but it is difficult to convey the idea in text or with pictures. You will want someone to professionally walk you through the whole ordeal if the above settings do not work. Your ISP will start with basic troubleshooting of your connection, likely by measuring signal strength coming off your modem, and by making you do a number of mundane tasks like turning things off and on. This is usually all they are trained to do. They may even reluctantly move on to a possible at-home support request. Most technicians who do at-home service for major ISPs in home environments either do not know how to configure a 3rd-party Cisco router, or refuse to do so, since it is not their priority (they are trained to wire cable and set up modems). As far as I know, however, Cisco's support unit, will still in fact walk you through a router set up over the phone provided you have the correct information from the ISP. Still be prepared for one support division to claim you have to call the other and vice versa.