When do equivariant quasi-isomorphisms of chain complexes induce a quasi isomorphism on the tensor product

When do equivariant quasi-isomorphisms of chain complexes induce a quasi
isomorphism on the tensor product

Suppose I have chain complexes $A,B,C,D$ where $A$ and $C$ have right
$R$-module structures and $B$ and $D$ have left $R$-module structures, and
that I have maps $f:A\to B$ and $g:C\to D$ which satisfy $g(r\cdot
x)=r\cdot g(x)$ and $f(x\cdot r)=f(x)\cdot r$. Suppose further that $f,g$
are quasi-isomorphisms.
Question: When can I conclude that $f\otimes_{R} g: A\otimes_{R} B\to
C\otimes_{R} D$ also induces a quasi-isomorphism?