We derive the TBA system of equations from the S-matrix describing integrable massive perturbation of the coset G_l×G_m/G_l+m by the field (1,1,adj) for all the infinite series of simple Lie algebras G=A_n,B_n,C_n,D_n. In the cases A_n,C_n, where the full S-matrices are known, the derivation is exact, while the B_n,D_n cases dictate some natural assumption about the form of the crossing-unitarizing prefactor for any two fundamental representations of the algebras. In all the cases the derived systems are transformed to the corresponding functional Y-systems and shown to have the correct high temperature (UV) asymptotic in the ground state, reproducing the correct central charge of the coset. Some specific particular cases of the considered S-matrices are discussed.