Experimentally Pierre Louis Dulong and Alexis Thérèse Petit had found in 1819 that the heat capacity per weight (the mass-specific heat capacity) for 13  measured elements was close to a constant value, after it had been multiplied by a number representing the presumed relative atomic weight of the element. These atomic weights had shortly before been suggested by John Dalton and modified by Jacob Berzelius.
Dulong and Petit were unaware of the relationship with R, since this constant had not yet been defined from the later kinetic theory of gases. The value of 3R is about 25 joules per kelvin, and Dulong and Petit essentially found that this was the heat capacity of certain solid elements per mole of atoms they contained.
The Kopp's law developed in 1865 by Hermann Franz Moritz Kopp extended the Dulong–Petit law to chemical compounds from further experimental data.
Amedeo Avogadro remarked in 1833 that the law did not fit the experimental data of carbon samples. In 1876, Heinrich Friedrich Weber, noticed that the specific heat of diamond was sensible to temperature.
In 1877, Ludwig Boltzmann showed that the constant value of Dulong–Petit law could be explained in terms of independent classical harmonic oscillators. With the advent of quantum mechanics, this assumption was refined by Weber's student, Albert Einstein in 1907, employing quantum harmonic oscillators to explain the experimentally observed decrease of the heat capacity at low temperatures in diamond.
Peter Debye followed in 1912 with a new model based on Max Plancks photon gas, where the vibrations are not to individual oscillators but as vibrational modes of the ionic lattice. Debyes model allowed to predict the behavior of the ionic heat capacity at temperature close to 0 kelvin, and as the Einstein solid, both recover the Dulong–Petit law at high temperature. 
The electronic heat capacity was overestimated by the 1900 Drude-Lorentz model to be half of the value predicted by Dulong–Petit. With the development of the quantum mechanical free electron model in 1927 by Arnold Sommerfeld the electronic contribution was found to be orders of magnitude smaller. This model explained why conductors and insulators have roughly the same heat capacity at large temperatures as it depends mostly on the lattice and not on the electronic properties.