We report temperature-dependent thermal-conductivity, κ, measurements on the layered quasi-two-dimensional organic superconductors κ-(BEDT-TTF)₂Cu(NCS)₂ and κ-(BEDT-TTF)₂Cu[N(CN)₂]Br down to 160 mK. The results for κ-(BEDT-TTF)₂Cu(NCS)₂ may be consistent with a nodal superconducting (SC) gap structure as indicated by a non-negligible remnant linear contribution when κ/T ∝ T² is extrapolated to T = 0. For κ-(BEDT-TTF)₂Cu[N(CN)₂]Br, contrary to expectations, higher κ values are observed in the superconducting regime as compared to the normal, high-field state evidencing a dominant phonon contribution to κ in the superconducting state. The strong increase of κ in the normal state below T _c for both samples indicates strong electron-phonon scattering. Our results highlight the need for thermal-conductivity measurements performed down to significantly lower temperatures to determine the symmetry of the SC gap.