Here's a rather obvious way to do it, in the case where all $C_i = 0$.

Let $X_1, \ldots, X_k$ be a basis of solutions of $A_1 X + X B_1 = 0$, so the general solution of $A_1 X + X B_1 = 0$ is $X = \sum_{i=1}^k t_i X_i$. For this to satisfy $A_2 X + X B_2 = 0$, we need $\sum_{i=1}^k t_i (A_2 X_i + X_i B_2) = 0$, which is a set of linear equations in the $t_i$. Solve and recurse...

Here's a rather obvious way to do it, in the case where all $C_i = 0$.

Let $X_1, \ldots, X_k$ be a basis of solutions of $A_1 X + X B_1 = 0$, so the general solution of $A_1 X + X B_1 = 0$ is $X = \sum_{i=1}^k t_i X_i$. For this to satisfy $A_2 X + X B_2 = 0$, we need $\sum_{i=1}^k t_i (A_2 X_i + X_i B_2) = 0$, which is a set of linear equations in the $t_i$. Solve and recurse...

Of course for this to be efficient, you'd want $k$ to be fairly small. In the worst case (e.g. $A_1 = B_1 = I$), $k = n^2$ where these are $n \times n$ matrices, but that's rather exceptional.