Fundamental understanding of the charge transport physics of hybrid lead halide perovskite semiconductors is important for advancing their use in high-performance optoelectronics. We use field-effect transistors (FETs) to probe the charge transport mechanism in thin films of methylammonium lead iodide (MAPbI3). We show that through optimization of thin-film microstructure and source-drain contact modifications, it is possible to significantly minimize instability and hysteresis in FET characteristics and demonstrate an electron field-effect mobility (mFET) of 0.5 cm2/Vs at room temperature. Temperature-dependent transport studies revealed a negative coefficient of mobility with three different temperature regimes. On the basis of electrical and spectroscopic studies, we attribute the three different regimes to transport limited by ion migration due to point defects associated with grain boundaries, polarization disorder of the MA+ cations, and thermal vibrations of the lead halide inorganic cages.