I had exactly the same issue, also starting just within the last week. When I navigated through the Apple TV menus to "Internet", and selected the first item, "Netflix", I got customized top-picks along the top. These were playable. And, they were accurate suggestions for my account - meaning that ATV was logging in successfully to and streaming from Netflix servers.
But when I tried to click on "Netflix" to enter the full Netflix application, I got an "Accessing Netflix" message, the spinning thing, and then nothing. No error message. And this message never timed-out, even when left overnight.
Nothing resolved this. I tried all the random Apple TV suggestions from other threads in this forum: power-cycling ATV, power-cycling router, ATV "Reset", changing DNS server, etc. I spent hours with Netflix Support, who were as confused as I was.
Finally, I did a Restore to factory settings (http://support.apple.com/kb/ht4367). It worked!
NetFlix streams without problems. But this wipes out all your customizations - anything you've done to your ATV box - and you'll have to re-do all that.
My Apple TV version was 4.4.4, same as Dui07. This is apparently a few versions old (http://support.apple.com/kb/ht4448). I'd been trying to do "Update Software" and ATV had said my software was up-to-date, but I guess I was encountering a separate issue which broke updates in v.4.4.4 (http://support.apple.com/kb/TS4107).
Netflix must have made some update this week that broke compatibility with older ATV software versions.