Probably the simplest method would just be to add a turn marker. Have an area on the board with the numbers one through five printed out on it, with an "It's Brazening Time!" highlight around the the number five.
If you're playing a game with a definite number of turns, just print all turn numbers with a highlight around every fifth. If you have an indefinite number of turns, just mark out a five turn cycle, with the turn marker going back to turn one after every Brazening.
This may not be ideal since it can be easy for players to forget to advance a turn marker during the heat of the game, but it's simple to figure out, requires only a single moving part (i.e., the turn marker itself) and is a common enough mechanic in a lot of games. This sort of system can be added to pretty much any game, regardless of mechanic; what it lacks in ideality it makes up for in universal compatibility.
A more ideal system would be to somehow tie it into an existing game mechanic (e.g., the above suggestion works much better if you're already tracking turns because of a definite turn limit). RoToRa gives some good examples of this in his answer, but exactly what will and will not work is very dependant on the mechanics of the game itself.
In your game, it would be well worth looking at any action that must be performed every turn, and find some way to map it to the five-turn cycle.