For a randomly moving particle. Or, I suppose that 1/3 could generalise to 1/n, where n is the non rotational degrees of freedom for that particle.
Related reference Kinetic Theory of Gasses.
My understanding is that this question is being asked in the context of the kinetic theory of classical gases. In that context, here is the argument:
If the system is rotationally invariant, then we should have $\langle v_x^2 \rangle = \langle v_y^2 \rangle = \langle v_z^2 \rangle$. Thus $\langle v^2 \rangle = \langle (v_x^2 + v_y^2 + v_z^2 )\rangle $ which gives $\langle v^2 \rangle = 3 \langle v_x^2 \rangle $. Your comment about generalization to n dimensions is also correct.