Abstract : The detection of diffuse radio emission associated with clusters of galaxies indicates populations of relativistic leptons infusing the intracluster medium. Those electrons and positrons are either injected into and accelerated directly in the intracluster medium, or produced as secondary pairs by cosmic-ray ions scattering on ambient protons. Radiation mechanisms involving the energetic leptons together with decay of neutral pions produced by hadronic interactions have the potential to produce abundant GeV photons. Here, we report on the search for GeV emission from clusters of galaxies using data collected by the Large Area Telescope (LAT) on the Fermi Gamma-ray Space Telescope (Fermi) from August 2008 to February 2010. Thirty-three galaxy clusters have been selected according to their proximity and high mass, X-ray flux and temperature, and indications of non-thermal activity for this study. We report upper limits on the photon flux in the range 0.2-100 GeV towards a sample of observed clusters (typical values 1-5 x 10^-9 ph cm^-2 s^-1) considering both point-like and spatially resolved models for the high-energy emission, and discuss how these results constrain the characteristics of energetic leptons and hadrons, and magnetic fields in the intracluster medium. The volume-averaged relativistic-hadron-to-thermal energy density ratio is found to be < 5-10% in several clusters.