The effect of deuteration on the thermal conductivity kappa of water, crystalline ice, and amorphous ices was studied using the pressure induced amorphization of hexagonal ice, ice Ih, to obtain the deuterated, D2O, forms of low-density amorphous (LDA), high-density amorphous (HDA), and very-high density amorphous (VHDA) ices. Upon deuteration, kappa of ice Ih decreases between 3% and 4% in the 100-270 K range at ambient pressure, but the effect diminishes on densification at 130 K and vanishes just prior to amorphization near 0.8 GPa. The unusual negative value of the isothermal density rho dependence of kappa for ice Ih, g = (d ln kappa/d ln rho)(T) = -4.4, is less so for deuterated ice: g = -3.8. In the case of the amorphous ices and liquid water, kappa of water decreases by 3.5% upon deuteration at ambient conditions, whereas K of HDA and VHDA ices instead increases by up to 5% for pressures up to 1.2 GPa at 130 K, despite HDA's and VHDA's structural similarities with water. The results are consistent with significant heat transport by librational modes in amorphous ices as well as water, and that deuteration increases phonon-phonon scattering in crystalline ice. Heat transport by librational modes is more pronounced in D2O than in H2O at low temperatures due to a deuteration-induced red-shift of librational mode frequencies. Moreover, the results show that kappa of deuterated LDA ice is 4% larger than that of normal LDA at 130 K, and both forms display an unusual temperature dependence of kappa, which is reminiscent of that for crystals (kappa similar to T (-1)), and a unique negative pressure dependence of kappa, which likely is linked to local-order structural similarities to ice Ih.