  We consider classical strings propagating in a background generated by a
sequence of TsT transformations. We describe a general procedure to derive the
Green-Schwarz action for strings. We show that the U(1) isometry variables of
the TsT-transformed background are related to the isometry variables of the
initial background in a universal way independent of the details of the
background. This allows us to prove that strings in the TsT-transformed
background are described by the Green-Schwarz action for strings in the initial
background subject to twisted boundary conditions. Our construction implies
that a TsT transformation preserves integrability properties of the string
sigma model. We discuss in detail type IIB strings propagating in the
\g_i-deformed AdS_5 x S^5 space-time, find the twisted boundary conditions for
bosons and fermions, and use them to write down an explicit expression for the
monodromy matrix. We also discuss string zero modes whose dynamics is governed
by a fermionicgeneralization of the integrable Neumann model.
