  In this paper we construct infinitely many Shimura curves contained in the
locus of Jacobians of genus four curves. All Jacobians in these families are
${\mathbb Z}/3$ covers of varying elliptic curves that appear in a geometric
construction of Pirola, and include an example of a Shimura-Teichmuller curve
that parameterizes Jacobians that are suitable ${\mathbb Z}/6$ covers of
${\mathbb P}^1$. We compute explicitly the period matrices of the Shimura
curves we construct using the original construction of Shimura for moduli
spaces of abelian varieties with automorphisms.
