It is worth pointing out that in a commutative ring, if $x$ is a zero divisor, then so is $xy$, unless $xy=0$. The reason is that, if there is some $z$ such that $xz=0$, then $(xy)z=(xz)y=0$. This means that the condition "$abc$ is not a zero divisor" means that $abc=0$. Further, since $ab, bc, ac$ are all nonzero, this means that $a,b,c$ are all zero divisors.
Since the way an element of $\mathbb Z^3$ is a zero divisor is if one of its coordinates is zero, what matters in an example is which coordinates are zero. One might ask "are there any examples that are not of the form "$(x,y,0),(z,0,w),(0,s,t)$" (as such examples are fundamentally relying off of the same key idea as the given example). In fact, there are not!
Between $a,b,c$, we need to have a zero in each of the three coordinates, and each term must have at least one zero. The question is, can we have an example where, say, $a=(1,0,0)$? The answer is no! Because $ab\neq 0$, we can't have $0$ in the first slot of $b$. Similarly, we couldn't have a zero in the first slot of $c$. But then the first slot of $abc$ has to be nonzero too (because $\mathbb Z$ has no zero-divisors).
So not only does the given example work, but it is essentially the only example, conceptually speaking.