The countdown is done automatically except when you click on "replay" in a game. But if you go back to the game index (without using the previous browser button) or refresh the page (F5), you can see how many Rounds you have left. Be careful, do not refresh in the middle of a Round otherwise the Round is lost.
I already proposed it a few years ago, but I think it would be nice to count the 15 free rounds. I almost always play the same game, and it would be nice to know where we stand. Of course, it is not an obligation, just a suggestion from one of your best followers.