  The recently constructed Lunin-Maldacena deformation of $AdS_{5}\times S^{5}$
is known to support two inequivalent Penrose limits that lead to BPS pp-wave
geometries. In this note, we construct new giant graviton solutions on these
backgrounds. A detailed study of the spectra of small fluctuations about these
solutions reveals a remarkably rich structure. In particular, the giants that
we contruct fall into two classes, one of which appears to remain stable in the
Penrose limit independently of the strength of the deformation. The other class
of giants, while more difficult to treat analytically, seems to exhibit a shape
deformation not unlike the so-called "squashed giants" seen in the pp-wave with
a constant NS $B$-field turned on. Some consideration is also given to the
associated giant operators in the BMN limit of the dual ${\cal N}=1$ SYM gauge
theory.
