Non-equilibrium molecular dynamics (NEMD) simulations are used to compute the phonon thermal conductivity (k) for liquids and glasses of composition Mg2SiO4, CaMgSi2O6, and NaAlSi3O8 at 2000–4500 K and 0–30 GPa based on classical potentials. These compositions span the range of melt polymerization states in natural systems at ambient pressure. The NEMD results compare well with available laboratory measurements on molten NaAlSi3O8 and CaMgSi2O6 at 1 bar. Thermal conductivities decrease with increasing temperature (T), increase with increasing pressure (P), and at low pressure increase slightly as the mean coordination number of Si and Al around oxygen increases, in the sequence Mg2SiO4, CaMgSi2O6, and NaAlSi3O8. At 3500 K, the thermal conductivity of CaMgSi2O6 at 0, 10, 20, and 30 GPa is 1.1, 2.1, 2.5, and 3 W/mK, respectively. At ambient pressure (0.2 ± 0.15 GPa), k = 1.2 and 0.5 W/mK at 2500 and 4500 K, respectively, for CaMgSi2O6. For NaAlSi3O8 composition, k varies from 1.7 to 2.7 W/mK at 3050 K for pressures of 6 and 30 GPa, respectively. Mg2SiO4 liquid at ambient pressure (0.07 ± 0.16 GPa) is found to have thermal conductivities of 1.36 and 0.7 W/mK at 2500 and 4500 K, respectively. Tables giving computed k values for all compositions are included for state points studied. The trade-off between T and P implies that the phonon thermal conductivity of silicate liquids at mantle depths increases substantially (factor of 2–3) along isentropes.