Q: How do you watch channels on Apple TV when you have discontinued cable service?

That's how Apple TV apps work. You must have an internet connection to view AMC, Comedy Central, etc.

I'm not sure that the "whole purpose of getting Apple TV is to nix the expensive cable service". For me, it's a way to watch Netflix, Hulu and iTunes purchases on a large screen. I haven't had cable for about 20 years, since long before the ATV came out.

Contact the networks that don't make subscription based apps available in the App Store and tell them you'd like them to. HBO and Showtime are examples of content providers that do allow you to subscribe to their content directly through apps.

At this time Apple TV "apps" for many cable television content providers will ask you to provide proof of subscription through your cable provider, not Apple. For those 'apps' (A&E, Fox Now, FX, etc.) If you don't have that you can't authorize yourself to use their content for free. These are often separate entities from the Netflix, HBO's and Hulu's which are "streaming services"who have a enabled this without the need for a cable subscription.

ATV apps that are proprietary to cable network program providers. Otherwise it would be "strange" to have to pay for music and movies on iTunes, or buying apps in the app store as well for content Apple doesn't own, it simply provides a store front for it.

Unfortunately, that's the way the content owners have decided to do things. It's not just ATV that's affected. You'll find that's true with most streaming media devices. In addition, even if you have a paid subscription to a service like Hulu, there are certain programs you can't watch unless you also subscribe to a cable or satellite provider with that channel in your package (SyFy for example).

For what it's worth, I jumped ship from U-Verse for TV to DirecTV. AT&T bought DirecTV specifically so that they could kill off their U-Verse TV service. It was never profitable and it was affecting their ability to deliver higher speed Internet services in the markets where it was available. That's why they stopped offering the promotional deals they used to have. I had dropped down to broadcast only service, like you, and had kept the DVR. I found that if I kept the AT&T Internet service (which I did), I could get DirecTV for a fixed price of $50 per month including all the equipment and DVR for 4 TVs for 2 years. That's only about $3 more than I was paying for broadcast only U-Verse.

The wife complains when the signal drops occasionally during really bad storms, but I just remind her we could just go back to broadcast only service if the likes... that usually ends the discussion pretty quickly.

Other than that, it's been great.  The cable companies in my area couldn't come close to matching the price for the package.