To sketch this piecewise function, let's break it down into its components based on the specified intervals:

1. For $x < -3$:
   The function is $f(x) = -x - 4$.
   This is a linear function with a slope of -1 and a y-intercept at -4.
   Since it’s defined for $x < -3$, we will graph it starting at $x = -3$, without including this point (an open circle at $x = -3$).

2. For $-3 \leq x \leq 3$:
   The function is $f(x) = x$.
   This is a linear function with a slope of 1, passing through the origin (0,0).
   We will graph this line segment from $x = -3$ to $x = 3$, including both endpoints (closed circles).

3. For $x > 3$:
   The function is $f(x) = (x - 4)^3$.
   This is a cubic function shifted 4 units to the right.
   We will start graphing it at $x = 3$ but not include $x = 3$ (open circle at $x = 3$).

Let me plot this graph for you.