Question: I'm working on a new board game where there's an action (we'll call it *Brazening*) that only happens every X turns (let's say it happens every 5 turns).

What's a good way to keep track of when it's time for the next *Brazening*?

The perfect method would:

* be easy for players to do
* be hard for players to forget
* require a minimum of pieces or moving parts
* not require pencil and paper

There might not be a perfect method.

Answer: 
If you only need the *Brazening* to happen *approximately* every X turns, you could simply roll a die at the beginning of every turn with a 1/X chance. For a 1/5 chance, for instance, roll a D10, and if it comes up 1 or 2 then *Brazen* away!

Whether or not this method is useful would depend completely on the purpose of the *Brazening* mechanic and the importance of its predictability. If it's necessary for some kind of restocking of important materials (like drawing cards or recovering health), this method probably won't work. On the other hand, if the purpose of the *Brazening* is to periodically make things more difficult/interesting for your players, or to temporarily modify other mechanics to mix things up, its randomness might be desirable!
Here is a solution for 3 players and more that only needs one object to move and only at
*Brazening! time* so it is unlikely to be forgotten.
I think it meets all your requirements and might blend in your game
without looking too artificial.

In its basic form the number of turns between *Brazening! times* depends
on the number of player which may, or may not be a good feature,
depending on your secret game mechanics.

The idea is that *Brazening!* happens when the player has the Brazen
token during his turn.
The *Brazen token* might be anything but making it substantially big
(such as the totem in Jungle Speed) decrease the likelihood of
forgetting it.
The *Brazen token* is passed from player to player at each *Brazening!
time* in the opposite sens of the change of turn.

The following drawing explains it for four players marked by numbered
circles.
The turn direction is indicated in orange and the *Brazen token* passing
direction is in blue.
The *Brazen token* is figured by the blue 'B' and the turn by the orange
'T'.
The number of the turn is written in orange in the centre square.

![Sequence of turns for four players](https://i.stack.imgur.com/zijeZ.png)

Based on this simple mechanic, you can adjust the number of turn
between *Brazening! times* by altering the passing rule.

If *Brazening!* requires some specific actions to be done, then you can
ask the player having the token to do it so the player doing the *Brazening!* changes every time. It might even been done *with* the object, thus making it blend smoothly with the game theme.