Computer won't recognize router unless I unplug/replug the router

I have been having a WEIRD problem on the older one of my two computers for the past few months. Most of the time, when I turn on my computer, there is NO Internet connection unless I unplug and replug the router. (This is a desktop computer directly connected to the router with a cable.)

If I go to Network Connections on my control panel right after I boot up, my Local Area Connection (Ethernet card) will say "connected", but the actual Internet Gateway will not even be listed at all. The unplug/replug will make it appear and connect, then no problems for several hours.

Before you say this is a router problem, I replaced the router about a month ago to upgrade to an N router, and the problem happened with both the old and new routers. Never had the problem for 2 or 3 years with the original router, then one day it just randomly started.

I also replaced the Ethernet card on this computer with a new one just in case that was the problem. Nope; same thing still happens.

As a final irony, my second computer (a newer one running a quad core) is plugged into the same router and never loses Internet connectivity ever. Neither do my Xbox, PS3, Wii, Tivo, wireless Skype phone or anty of the other half-dozen wireless devices in my house.

Ugggggghhh.... I really do not want to reinstall Windows if I can help it -- probably like most people that have a 5-year-old computer, I know it would make all my problems go away, but I have so much stuff that I don't want to lose, that doing that is a last last last last resort.

However, I am 100% sure that the "allow the computer to turn off this device to save power" option IS checked. When I installed the new ethernet card, a menu box came up and that was what it was set to by default, and the box was checked. Stupidly, I didn't think it would affect anything, so I didn't change it.

I will go try changing that option and see if it helps.

Thanks --- hopefully!