  We determine Grothendieck groups of periodic derived categories. In
particular, we prove that the Grothendieck group of the $m$-periodic derived
category of finitely generated modules over an Artin algebra is a free
$\mathbb{Z}$-module if $m$ is even but an $\mathbb{F}_2$-vector space if $m$ is
odd. Its rank is equal to the number of isomorphism classes of simple modules
in both cases. As an application, we prove that the number of non-isomorphic
summands of a strict periodic tilting object $T$, which was introduced in [S21]
as a periodic analogue of tilting objects, is independent of the choice of $T$.
