We compute the periods of compactifications of complements of a special class of hyperplane arrangements. In particular, we exhibit a combinatorial condition on the intersection lattice of a hyperplane arrangement that ensures that its associated periods are linear combinations of special values of multiple polylogarithms. Our method generalizes Brown's strategy to compute the periods of moduli spaces of curves of genus zero.

We apply this result to the reflection arrangements of full monomial groups. We show that the periods of their corresponding compactifications are linear combinations of values of multiple polylogarithms at roots of unity.

Finally, we explicitly compute in terms of multiple zeta values a family of Brown's basic cellular integrals on the moduli space of stable curves of genus zero with marked points.