Thanks for all the great info, but the problem turned out to be my Lingo VOIP router..
I entered the OpenDNS.com IP addresses as the DNS servers for my Replay, its wireless access point, and my Netgear router - but the Replay still timed out setting its clock. On a whim I set the router to forward TCP/UDP port 123 to the Replay's IP, but that didn't work either.
My Netgear router connects to a Lingo VOIP router, which connects to my cable modem. In the VOIP router's config I have the Netgear router set as its DMZ server. However, somehow the VOIP router is blocking the Replay's clock update.
When I took the VOIP router out of the loop & plugged the Netgear router directly into the cable modem and rebooted everything, the Replay clock updated fine. Then I put things back the way they were & rebooted, and the Replay clock udpate timed out.
I thought setting the Netgear router as the VOIP's DMZ server would cause all non-VOIP traffic to be sent to the Netgear router, but apparently not..
So I'll just have to go through this process every few months to make sure my Replay clock stays updated. Which is slightly annoying but not a big hassle..
Thanks again for all the help!