  In this paper, we study the rainbow Erd\H{o}s-Rothschild problem with respect
to 3-term arithmetic progressions. We obtain the asymptotic number of
$r$-colorings of $[n]$ without rainbow 3-term arithmetic progressions, and we
show that the typical colorings with this property are 2-colorings. We also
prove that $[n]$ attains the maximum number of rainbow 3-term arithmetic
progression-free $r$-colorings among all subsets of $[n]$. Moreover, the exact
number of rainbow 3-term arithmetic progression-free $r$-colorings of
$\mathbb{Z}_p$ is obtained, where $p$ is any prime and $\mathbb{Z}_p$ is the
cyclic group of order $p$.

Combinatorics