We developed a full-band Monte Carlo simulator to investigate electron transport in a single layer of graphite (graphene). The electron and phonon dispersion curves of graphene are first obtained by applying the tight-binding method to the two inequivalent atoms of the graphene unit cell, considering their nearest four neighbors. The electron-phonon scattering rates for interactions that conserve momentum and energy are then determined using Fermi's golden rule and the deformation potential approximation. Average electron velocities due to external fields applied in different directions are calculated using these electron-phonon scattering rates and a semiclassical electron and electric field interaction. Graphene transport simulations at room temperature indicate large low-field graphene electron mobilities of approximately 4.0×104, 6.0×104, and 8.0×104 cm2/Vs along the ky (kx=0), kx=ky, and kx (ky=0) momentum directions, respectively. These low-field electron mobilities further suggest graphene field-effect mobilities of 1.3-2.6×104 cm2/Vs, which agree with experiments. Also, a large peak electron velocity of roughly 4.6×107 cm2/Vs at high fields is obtained.