Obviously if $\mathbf{A}=\begin{bmatrix}\mathbf{C} & \mathbf{0} \\ \mathbf{0} & \mathbf{D}\end{bmatrix}$ then $e^{A}=\begin{bmatrix}\mathbf{e^C} & \mathbf{0} \\ \mathbf{0} & \mathbf{e^D}\end{bmatrix}$.
Given any invertable matrix X we can express it as $X=e^A$ (but complex matrix A is not unique until X is not from one-parameter subgroup of GL(n,$\mathbb{C}$)). Anyway: if given X has a block diagonal structure then A has to have block diagonal structure as well?