  We consider the correspondence between the spinning string solutions in
Lunin-Maldacena background and the single trace operators in the
Leigh-Strassler deformation of N=4 SYM. By imposing an appropriate rotating
string ans\"atz on the Landau-Lifshitz reduced sigma model in the deformed
SU(2) sector, we find two types of `elliptic' solutions with two spins, which
turn out to be the solutions associated with the Neumann-Rosochatius system. We
then calculate the string energies as functions of spins, and obtain their
explicit forms in terms of a set of moduli parameters. On the deformed
spin-chain side, we explicitly compute the one-loop anomalous dimensions of the
gauge theory operators dual to each of the two types of spinning string
solutions, extending and complementing the results of hep-th/0511164. Moreover,
we propose explicit ans\"atze on how the locations of the Bethe strings are
affected due to the deformation, with several supports from the string side.
