jmc

If an integer is congruent to 1 mod 3, then it can be written in the form $3k+1$. Therefore, we have two inequalities: $3k+1\ge 10$, and $3k+1\le 99$. The inequality $3k+1\ge 10$ has solution $k\ge 3$, and the inequality $3k+1\le 99$ has solution $k\le 32\frac{2}{3}$. Therefore, $k$ must be an integer between $3$ and $32$. There are $\boxed{30}$ such integers.