Metals conduct heat well because their electrons move easily, and this is more efficient than phonons in most other materials.
Though, diamond, silicon, graphite, alumina... can be excellent heat conductors, and some metals (titanium alloys) poor ones.
As electrons carry kT (or 3/2kT or whatever) as well as q, metals show a constant ratio between their heat and electricity conductances - at least under usual conditions. Useful if you lack data.
This has limits. For instance, superconductors carry heat badly.