Dijet production by almost real photons has been studied at HERA using the ZEUS detector. The photoproduction of dijets can be used to constrain the parton distributions in the photon and the proton and test perturbative QCD. Jets have been identified using two different cone algorithms and one cluster algorithm. Some comparative study of these algorithms was made. A cut on xyOBS, the fraction of the photon energy participating in the production of the two jets of highest transverse energy, is used to define and separate resolved and direct processes. Cross sections of these two types of process, and also for the full range of have been measured for dijets with transverse energies above 6, 8, 11, and 15 GeV. The cross sections are corrected for detector acceptance and compared to next to leading order QCD calculations. Agreement with these calculations is seen for the pseudorapidity dependence of the direct photon events with ETjet > 6 GeV and of the resolved photon events with ETjet > 11 GeV. Calculated cross sections for resolved photon processes with 6 GeV < ETjet < 11 GeV lie below the data.