  We study periodic wind-tree models, billiards in the plane endowed with
$\mathbb{Z}^2$-periodically located identical connected symmetric right-angled
obstacles. We exhibit effective asymptotic formulas for the number of (isotopy
classes of) periodic billiard trajectories (up to $\mathbb{Z}^2$-translations)
on Veech wind-tree billiards, that is, wind-tree billiards whose underlying
compact translation surfaces are Veech surfaces. We show that the error term
depends on spectral properties of the Veech group and give explicit estimates
in the case when obstacles are squares of side length $1/2$.
