  We study renormalization group flows among N=1 SCFTs realized on the
worldvolume of D3-branes probing toric Calabi-Yau singularities, thus admitting
a brane tiling description. The flows are triggered by masses for adjoint or
vector-like pairs of bifundamentals and are generalizations of the
Klebanov-Witten construction of the N=1 theory for the conifold starting from
the N=2 theory for the C^2/Z_2 orbifold. In order to preserve the toric
condition pairs of masses with opposite signs have to be switched on. We offer
a geometric interpretation of the flows as complex deformations of the
Calabi-Yau singularity preserving the toric condition. For orbifolds, we
support this interpretation by an explicit string amplitude computation of the
gauge invariant mass terms generated by imaginary self-dual 3-form fluxes in
the twisted sector. In agreement with the holographic a-theorem, the volume of
the Sasaki-Einstein 5-base of the Calabi-Yau cone always increases along the
flow.
