Our Apple TV picture quality has recently become grainy and looks like an old movie.

Please try the following:

1.     Unplug your modem from its power source and Restart Modem.

2.     Unplug your router from its power source and Restart router.

3.     Unplug Apple TV from Power for 2-3 minutes – re-plug in and retest. 

If those steps do not work, please try another HDMI cable.

PLEASE re-read my first post, you did not follow all the steps. You may have completed one step but you don't say how you restarted. Using the restart command and unplugging from power are 2 separate actions that can have different results.