How do I turn 1 objects rotation into another objects movement direction?

I’m making a VR project. The VR Camera Rig contains 2 controllers, which are rotated freely by the player. I want to apply a continuous movement speed to the camera rig that can be adjusted with inputs, which is pretty easy and straight-forward.
But the part I’m having trouble wrapping my head around is this: I want the Camera Rig to change movement direction -but NOT rotate- in the direction that one of the controllers is pointing.
IE: The Camera Rig and I are facing forward, and if I point the controller to the bottom-left, the Camera rig will move that (or any) direction but continue to face forward.
I feel like the answer is obvious, and that maybe this has revealed a gap in my understanding of simple coding principals. Either way, I would appreciate any suggestions or explanations.

You can attach Something like this to the some component in the rig inside Update:

this.transform.Translate(controllerTransform.forward * speed * Time.deltaTime);

or

this.transform.position = Vector3.MoveTowards(this.transform.position, controllerTransform.forward + this.transform.position, speed * Time.deltaTime);

where controllerTransform is the Transform component of you controller and speed is float value representing the speed in units per second.