The answer to your question is very wide and include many phenomena. There is no single mechanism that converts absorbed energy into heat. I will give you a general overview on this topic.
Heat is transferred by radiation which is in general an electromagnetic wave (not light only). For example, IR radiation alone provides 49% of the heat provided to earth from sun radiation. Have a look at Heat section of this page.
In electromagnetic spectrum, the wavelength varies significantly, which means that materials respond differently to different ranges of wavelengths. For example (listing from long wavelength to short wavelength):
Radio waves: The wavelength here is very long such that the materials constituents (atoms and molecules) don't sense the waves. Thus most solids are transparent to radio waves, which means there is no wave-material interaction. There is no heat generated.The transparency is clear as you can get cellphone reception in your home as the wave simply travel through walls and our bodies.
Microwaves: The wavelength here is shorter, the photons energy in this level are comparable to the energy levels of the molecular rotational levels, which means that a molecule could absorb a microwave photon and start to rotate. This rotation is a form of kinetic energy of molecules which is translated as heat when you look at the whole ensemble of molecules. This mechanism is the mechanism known for microwave heating in your kitchen.
IR: The photon energy in this range is comparable to molecular vibrational levels. So if a molecule absorbed an IR photon it will vibrate, which is translated into heat when you look at the whole ensemble of molecules.
Visible light: things get complicated here, the photon energy here is comparable to electronic levels within an atom. There is no simple direct explanation of how photon energy is transformed into heat.
One can argue for visible light that an electron can absorb a photon and radiate it without increasing kinetic energy of atoms (no increase oh heat). That is true but that is a very simplistic picture of what happens in reality. First because the visible light is continuous spectrum, while the photons that can be absorbed by electrons are discrete. So there are many visible light photons with wavelengths that electrons can't absorb, which could be absorbed by ions or affect polar bonds somehow if they existed in the material. Second, some materials like solids (where heat by radiation is efficient) have energy bands rather than the simplistic electron levels concept. So the reaction of materials to visible light in this case is complex. But one of the mechanism of transforming photon energy to heat that I can think of here is the absorption of visible light by ion lattice in some crystals.
UV: The photons in this range have enough energy to ionize the atoms they hit (by liberating an electron from its orbital). Again, there is no simple direct explanation of how photon energy is transformed into heat. One possible route is the inelastic scattering of electrons with materials atoms. For example, have a look at page 5 of this report where it is mentioned that inelastic electron scattering can induce phonons (lattice vibration, which is a form of kinetic energy of atoms)
X rays: The photon energy in this range is much larger than the energy required to move electron from one level to other. X rays photons can ionize an atom and become lower energy photon through Compton scattering. There is no simple direct explanation of how photon energy is transformed into heat.
Briefly, heating by radiation in general can't be attributed to a simple explanation. The mechanisms through which photons energy is transformed into heat depend on the energy of the photon and the properties of the material. The simplistic picture of a photon being absorbed by electron is narrow to explain conversion to heat because it only describes a single electron in a free atom. That is a narrow view of material properties.
Have a look here and the nice figure of Non-ionizing radiation section of this page
Hopefully that was helpful