We consider D3 branes at orbifolded conifold singularities which are not quotient singularities. We use toric geometry to study the moduli space of the gauge theories on the D3 branes. We find that topologically distinct phases are related by a flop transition. It is also shown how the conifold singularity is partially solved to an orbifold singularity in some phases if we give expectation values to some of the chiral fields.