How do you lock the first-person CharacterController to only walk along the x-axis?

You should modify the CharacterController movement script. If you're using the standard First Person Controller, open the CharacterMotor.js script and find t$$anonymous$$s instruction:

       // Move our character!
    movement.collisionFlags = controller.Move (currentMovementOffset);

Add one instruction to zero the z movement before Move:

       // Move our character!
    currentMovementOffset.z = 0; //<- zero movement in the z direction
    movement.collisionFlags = controller.Move (currentMovementOffset);

But be aware that CharacterMotor.js is a big monster that eats experienced programmers in the breakfast! Most modifications only result in undesired collateral effects, thus t$$anonymous$$s one may also have weird consequences!

You would be interested in the "Constraints" section of the "Rigidbody" voice, in the Inspector. Look at here: http://unity3d.com/support/documentation/Components/class-Rigidbody.html. Search for the voice "Freeze position", and make sure that you've only the check on Z-axis, deactivating X- and Y- ones. However, t$$anonymous$$s works for physics, and for forces related to it: if you have a script that move the character using WASD, or arrows, movements along all these direction still remain valid. Reading at your question, it seems that you are using the "3d person controller", that's implemented by default on Unity. My advice is to make a script of yours, very easy indeed. Look at t$$anonymous$$s:

 var speed = 20.0;
 
 function Update () {
    var x = Input.GetAxis("Horizontal") * Time.deltaTime * speed;
    var y = Input.GetAxis("Vertical") * Time.deltaTime * speed;
    transform.Translate(x, y, 0);
 }

The script above make the spaces$$anonymous$$p controllable only on X- and Y- axis, using arrows or WASD. Hope that it helps.