We can go straight from your definition of a zero-divisor. In $A \times B$, a zero divisor is any two non-zero elements that multiply to give zero. Note that in this new ring, "zero" is the element $(0, 0)$.
So the elements $(a, b)$ and $(c, d)$ are a zero divisor pair if $(a, b) \cdot (c, d) = (0, 0)$.
If $A$ or $B$ originally had zero divisors, then an easy way to generate zero divisors for $A \times B$ is as follows. Say $a, c \in A$ is a zero divisor pair and $b, d \in B$ is also a zero divisor pair. Then the following are zero divisor pairs in $A \times B$:
$$(a, 0)(c, 0) = (0, 0)$$
$$(a, b)(c, d) = (0, 0)$$
$$(0, b)(0, d) = (0, 0)$$
Furthermore, given the way multiplication has been defined in $A \times B$, any elements of the form $(x, 0)$ and $(0, y)$ are also zero-divisors.