We present an ab initio approach to the solar coronal heating problem by modeling a small part of the solar corona in a computational box using a three-dimensional MHD code including realistic physics. The observed solar granular velocity pattern and its amplitude and vorticity power spectra, as reproduced by a weighted Voronoi tessellation method, are used as a boundary condition that generates a Poynting flux in the presence of a magnetic field. The initial magnetic field is a potential extrapolation of a SOHO/MDI high-resolution magnetogram, and a standard stratified atmosphere is used as a thermal initial condition. Except for the chromospheric temperature structure, which is kept nearly fixed, the initial conditions are quickly forgotten because the included Spitzer conductivity and radiative cooling function have typical timescales much shorter than the time span of the simulation. After a short initial start-up period, the magnetic field is able to dissipate (3-4)×106ergscm-2s-1 in a highly intermittent corona, maintaining an average temperature of ~106 K, at coronal density values for which simulated images of the TRACE 171 and 195 Å passbands reproduce observed photon count rates.