Most solid components I need to consider at work have a specific heat of about 1 kJ/kg/K.
Why is that so?
Any explanation as simple as for perfect gases?
Yes, it's mode-counting, similar to perfect gasses. Counting the vibrational modes of each atom in the lattice it's 3/2 kT for the kinetic modes and another 3/2 kT for the potential energies.
One of the solids in my long list is 3(CaO).SiO2 and another is 3(CaO)(MgO)2(SiO2).
This would mean that "heavier" molecules are not very rigid, therefore making room for more modes and more specific heat per molecule. The end result around 1 kJ/kg/K depends also on the atoms involved.
This makes more sense when you express it as 25 J/mol/K. Most solids have that as their specific heat. This is called the Law of Dulong and Petit. It works for exactly the reason Conway said.