Let the first odd integer be $x$. Then the next two consecutive odd integers are $x+2$ and $x+4$. The sum of these three integers is $$x+(x+2)+(x+4)=285.$$ Simplifying the left side gives $3x+6=285$. Subtracting 6 from both sides gives $3x=279$, so $x=93$. Therefore the three integers are $\boxed{93, 95, \text{ and } 97}$.
