The traction is defined as
$$
\mathbf{t} = \mathbf{n} \cdot \boldsymbol{\sigma}
$$
In terms of components, the zero-traction condition is
$$
t_j = \sum_i n_i \sigma_{ij} = 0
$$
From the above you can see that the stress components don't necessarily have to be zero for the traction to be zero.
For a linear elastic material,
$$
\boldsymbol{\sigma} = \mathsf{C}:\nabla{\mathbf{u}}
$$
In component form,
$$
\sigma_{ij} = \sum_k \sum_l C_{ijkl} \frac{\partial u_k}{\partial x_l}
$$
Once again, since there is no requirement that all the $C_{ijkl}$ values have to be positive, you don't need all the displacement gradients to be zero for the stresses to be zero (which is not strictly necessary for zero-tractions anyway).
However, zero displacement gradients will lead to zero tractions.