Find elements $a,b$, and $c$ in the ring $\mathbb Z \oplus\mathbb Z \oplus\mathbb Z $ such that $ab$, $ac$, and $bc$ are zero divisors but $abc$ is not a zero divisor.
I am not sure how to approach this problem guidance is appreciated.
The zero divisors of this ring are the elements that are zero in at least one place.
The product of two elements always has at least as many zeros as the factors. If the product of two elements has a zero then, a product of three elements certainly will have more.
Your example of $(1,1,0),(0,1,1),(1,0,1)$ will work provided you define away $0$ from being a zero divisor.