In an electromagnetic wave, the electric field ($\mathbf{E}$), magnetic field ($\mathbf{B}$), and the direction of propagation ($\mathbf{k}$) are mutually perpendicular. The direction of propagation is given by the direction of the cross product $\mathbf{E} \times \mathbf{B}$.

Given that the wave propagates in the $x$-direction ($\hat{i}$), we must satisfy two conditions:

1. Orthogonality: $\mathbf{E} \cdot \mathbf{B} = 0$ (Fields must be perpendicular).
2. Direction: $\mathbf{E} \times \mathbf{B}$ must be parallel to $\hat{i}$.

Testing Option D: Let $\mathbf{E} = -\hat{j} + \hat{k}$ and $\mathbf{B} = -\hat{j} - \hat{k}$.

1. Dot Product: $\mathbf{E} \cdot \mathbf{B} = (-1)(-1) + (1)(-1) = 1 - 1 = 0$. The fields are perpendicular.

2. Cross Product: $\mathbf{E} \times \mathbf{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\0 & -1 & 1 \\0 & -1 & -1 \end{vmatrix}$ $= \hat{i}((-1)(-1) - (1)(-1)) - \hat{j}(0) + \hat{k}(0)$ $= \hat{i}(1 + 1) = 2\hat{i}$. The result is along the $+\hat{i}$ direction ($x$-direction).

Other options fail either the orthogonality check (dot product $\neq 0$) or the direction check.