  We study a class of near-BPS operators for a complex 2-parameter family of
N=1 superconformal Yang-Mills theories that can be obtained by a
Leigh-Strassler deformation of N=4 SYM theory. We identify these operators in
the large N and large R-charge limit and compute their exact scaling dimensions
using N=1 superspace methods. From these scaling dimensions we attempt to
reverse-engineer the light-cone worldsheet theory that describes string
propagation on the Penrose limit of the dual geometry.
