  In this paper we study the N=1 supersymmetric field theories realized on the
world-volume of type IIB D3-branes sitting at orientifolds of non-orbifold
singularities (conifold and generalizations). Several chiral models belong to
this family of theories. These field theories have a T-dual realization in
terms of type IIA configurations of relatively rotated NS fivebranes, D4-branes
and orientifold six-planes, with a compact $x^6$ direction, along which the
D4-branes have finite extent. We compute the spectrum on the D3-branes directly
in the type IIB picture and match the resulting field theories with those
obtained in the type IIA setup, thus providing a non-trivial check of this
T-duality. Since the usual techniques to compute the spectrum of the model and
check the cancellation of tadpoles, cannot be applied to the case orientifolds
of non-orbifold singularities, we use a different approach, and construct the
models by partially blowing-up orientifolds of C^3/(Z_2 x Z_2) and C^3/(Z_2 x
Z_3) orbifolds.
