  A class of exactly solvable string models can be obtained by starting with
flat space and combining T-duality and shifts of angular coordinates of several
polar planes. The models are the analog of the Lunin-Maldacena
\beta-deformation of the AdS_5 x S^5 type IIB string background, which is dual
to a Leigh-Strassler deformation of \N=4 Super Yang-Mills Theory.
  We determine the complete physical string spectrum for two string models
obtained in this way, by explicitly solving the string equations and quantizing
in terms of free creation and annihilation operators. We also show that the
3-parameter (b_1,b_2,b_3) model, obtained by three independent TsT
transformations, has tachyons in some regions of the parameter space.
