The size of the automorphism group of a compact Riemann surface of genus g > 1 is bounded by 84(g-1). Curves with automorphism group of size this bound are called Hurwitz curves. In many cases the automorphism group of these curves is the projective special linear group PSL(2,q). We present a decomposition of the Jacobian varieties for all curves of this type and prove that no such Jacobian variety is simple.