The resistive tearing-mode instability of tangential-discontinuity current sheets is studied within the framework of linear double-polytropic MHD theory. Four sets of polytropic exponents, γ∥ and γ⊥, are chosen to describe various thermodynamic states. It is shown that the dependence of tearing growth rate on the energy equations, plasma beta as well as the magnetic By component is much more pronounced than that in isotropic plasmas; in particular, the growth rate is larger for smaller γ∥ and γ⊥ as well as for larger p⊥/p∥ and β⊥. For certain parameter regime, the growth rate may even increase with increasing magnetic Reynolds number, a result in contrast to the pure tearing instability. Cases of large growth rate on the order of Alfvén time scale are associated with oscillatory slow-mode structures that may sometimes exhibit positive density-magnetic field correlation.