The heating of the plasma confined in active regions of the solar corona is caused by the dissipation of magnetic stresses induced by the photospheric motions of the loop footpoints. The aim of the present paper is to analyze whether solar coronal heating is dominated by slow (DC) or rapid (AC) photospheric driving motions. We describe the dynamics of a coronal loop through the reduced magnetohydrodynamic equations and assume a fully turbulent state in the coronal plasma. The boundary condition for these equations is the subphotospheric velocity field that stresses the magnetic field lines, thus replenishing the magnetic energy that is continuously being dissipated inside the corona. In a turbulent scenario, energy is efficiently transferred by a direct cascade to the microscale, where viscous and Joule dissipation take place. Therefore, for the macroscopic dynamics of the fields, the net effect of turbulence is to produce a dramatic enhancement of the dissipation rate. This effect of the microscale on the macroscale is modeled through effective dissipation coefficients much larger than the molecular ones. We consistently integrate the large-scale evolution of a coronal loop and compute the effective dissipation coefficients by applying a closure model (the eddy-damped, quasi-normal Markovian approximation). For broadband power-law photospheric power spectra, the heating of coronal loops is DC dominated. Nonetheless, a better knowledge of the photospheric power spectrum as a function of both frequency and wavenumber will allow for more accurate predictions of the heating rate from this simple model.