Yes, there is a nice way. The mapping cone $C_i^*$ is the homotopy pushout of $\ast \gets A_i^* \to B_i^*$, and $-\otimes X$ is a monoidal left Quillen functor (symmetric if the underlying ring is commutative), so $-\otimes^L -$ commutes with homotopy colimits. Hence, $C_1^* \otimes^L C_2^*$ is the mapping cone of $A_1^* \otimes^L A_2^* \to B_1^* \otimes^L B_2^*$ . Derived functors and homotopy colimits can be computed via projective resolutions, so you have very classical inductive formulas to compute these quantities if you like.
EDIT: The original answer is too simplistic, so this edit fixes it (thanks Thomas for pointing this out). Instead of identifying $C_1^* \otimes^L C_2^*$ as "the mapping cone", it should say "$C_1^* \otimes^L C_2^*$ is the homotopy colimit of a cube built from the maps $A_i^* \to B_i^*$". My original answer was ignoring the grading, because I had just posted an answer, with the same "homotopy colimit" idea, to a simpler question.