  In general, the field equations of $f(R)$ theory coupled to a matter field
are very complicated and hence it is not easy to find exact analytical
solutions. However, if one considers traceless energy-momentum tensor for the
matter source as well as constant scalar curvature, one can derive some exact
analytical solutions from $f(R)$ theory coupled to a matter field. In this
paper, by assuming constant curvature scalar, we construct a class of charged
rotating black string solutions in $f(R)$-Maxwell theory. We study the physical
properties and obtain the conserved quantities of the solutions. The conserved
and thermodynamic quantities computed here depend on function $f'(R_{0})$ and
differ completely from those of Einstein theory in AdS spaces. Besides, unlike
Einstein gravity, the entropy does not obey the area law. We also investigate
the validity of the first law of thermodynamics as well as the stability
analysis in the canonical ensemble, and show that the black string solutions
are always thermodynamically stable in $f(R)$-Maxwell theory with constant
curvature scalar. Finally, we extend the study to the case where the Ricci
scalar is not a constant and in particular $R=R(r)$. In this case, by using the
Lagrangian multipliers method, we derive an analytical black string solution
from $f(R)$ gravity and reconstructed the function $R(r)$. We find that this
class of solutions has an additional logaritmic term in the metric function
which incorporates the effect of the $f(R)$ theory in the solutions.
