Thanks for the reply, Eunice.
I checked and found that my TV has a sleep timer too. Unfortunately, it only works with the built-in FireTV streaming function, so I had to switch to using that instead. It seems pretty similar to the Roku, so that's o.k. I just had to install all my apps again.
Problem solved for me, but I guess anyone who doesn't have Roku or Fire built into their TV is out of luck. They'll have to go back to using their clock radio.