Instead of computing this probability directly, we can find the probability that the product is odd, and subtract that from $1$.
The product of two integers is odd if and only if each of the two integers is odd.  The probability the first spinner yields an odd number is $2/3$ and the probability the second spinner yields an odd number is $1/3$, so the probability both yield an odd number is $(2/3)(1/3)=2/9$.
The desired probability is thus $1-2/9=7/9\rightarrow \boxed{\text{D}}$.
Each spinner is divided into $3$ equal parts.  The results obtained from spinning the two spinners are multiplied.  What is the probability that this product is an even number?

$\text{(A)}\ \frac{1}{3} \qquad \text{(B)}\ \frac{1}{2} \qquad \text{(C)}\ \frac{2}{3} \qquad \text{(D)}\ \frac{7}{9} \qquad \text{(E)}\ 1$