Thanks, Furry. The fact that it's only happening with this new router, and that it's so consistent and reproducible, doesn't seem like an ISP issue. More likely it's some sort of resource exhaustion issue. There's a whole burst of network traffic generated when the computer wakes up and other devices connect, and I suspect there's something in the router - a NAT table? A connection table? that is getting filled up. But, that's mostly intuitive speculation.
At any rate, the fact that I filled out the tech support contact form at D-Link and haven't had an acknowledgement or response is a deal-breaker for me. I've ordered a different router and arranged to return the DIR-857.
I appreciate all your help and suggestions! It seems like you are the major problem-solver around here!