  We consider N=1, D=4 superconformal U(N)^{pq} Yang-Mills theories dual to
AdS_5xS^5/Z_pxZ_q orbifolds. We construct the dilatation operator of this
superconformal gauge theory at one-loop planar level. We demonstrate that a
specific sector of this dilatation operator can be thought of as the transfer
matrix for a two-dimensional statistical mechanical system, related to an
integrable SU(3) anti-ferromagnetic spin chain system, which in turn is
equivalent to a 2+1-dimensional string theory where the spatial slices are
discretized on a triangular lattice. This is an extension of the SO(6) spin
chain picture of N=4 super Yang-Mills theory. We comment on the integrability
of this N=1 gauge theory and hence the corresponding three-dimensional
statistical mechanical system, its connection to three-dimensional lattice
gauge theories, extensions to six-dimensional string theories, AdS/CFT type
dualities and finally their construction via orbifolds and brane-box models. In
the process we discover a new class of almost-BPS BMN type operators with large
engineering dimensions but controllably small anomalous corrections.
