Following Gianluca Gorni's suggestion:
disttheta = TransformedDistribution[ArcCos[2*v - 1], v \[Distributed] UniformDistribution[{0, 1}]]
distr = TransformedDistribution[v^(1/3), v \[Distributed] UniformDistribution[{0, 1}]]
distphi = TransformedDistribution[v, v \[Distributed] UniformDistribution[{0, 2*Pi}]]
distz = TransformedDistribution[u*Cos[v], {u \[Distributed] distr, v \[Distributed] disttheta}]
PDF[distz, z]
