smc

Let $\underline{ABC}$ be one such odd positive $3$-digit integer with hundreds digit $A,$ tens digit $B,$ and ones digit $C.$ Since $\underline{ABC}\equiv 0(\mathrm{mod}3),$ we need $A+B+C\equiv 0(\mathrm{mod}3)$ by the divisibility rule for $3.$ As $A\in \{1,2,4,5,6,7,8,9\}$ and $C\in \{1,5,7,9\},$ there are $8$ possibilities for $A$ and $4$ possibilities for $C.$ Note that each ordered pair $(A,C)$ determines the value of $B$ modulo $3,$ so $B$ can be any element in one of the sets $\{0,6,9\},\{1,4,7\},$ or $\{2,5,8\}.$ We conclude that there are always $3$ possibilities for $B.$ By the Multiplication Principle, the answer is $8\cdot 4\cdot 3=\boxed{96}.$