BTW, totally, totally wrong. The first time the router handshakes with the ISP, it gets the (usually 2) DNS server addresses that the ISP uses. And every time you reset it after that (most router menus have a "reget the connection" button somewhere in one of the menus) it gets the DNS server IP(s) again. (Most of the time it also gets them when the DHCP lease expires.)
But using Google DNS or OpenDNS is probably the best way to start diagnosing this problem. If the ISP thinks of quality service as an expense, it may take them a while to notice that their DNS servers went down.
Then again, the cache in the router might, if left on long enough, become corrupted. (Although I've had SOHO Linkys run for 10 years without so much as looking at the case to see if it was still there. Not bad for $40, but if you take a look at their salary ranges, you'd better be sitting down. "You listed the Chief of Operations as a programmer.")