Summary: Motivated by the success of the recently proposed method of anomaly cancellation to derive Hawking fluxes from black hole horizons of spacetimes in various dimensions, we further extend the covariant anomaly cancellation method recently simplified by R. Banerjee and S. Kulkarni [Phys. Lett., B 659, No. 4, 827–831 (2008; Zbl 1246.83096)] to explore the Hawking radiation of the \((3+1)\)-dimensional charged rotating black strings and their higher-dimensional extensions in anti-de Sitter spacetimes, whose horizons are not spherical but can be toroidal, cylindrical or planar, according to their global identifications. It should be emphasized that our analysis presented here is very general in the sense that the determinant of the reduced \((1+1)\)-dimensional effective metric from these black strings need not be equal to one \((\sqrt{-g}\neq 1)\). Our results indicate that the gauge and energy-momentum fluxes needed to cancel the \({(1+1)}\)-dimensional covariant gauge and gravitational anomalies are compatible with the Hawking fluxes. In addition, the thermodynamics of these black strings is studied in the case of a variable cosmological constant.