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Weakly Interacting Massive Particles (WIMPs) are a theoretical class of particles that are excellent dark matter candidates. WIMP annihilation or decay may produce essentially monochromatic gamma rays detectable by the Fermi Large Area Telescope (LAT) against the astrophysical gamma-ray emission of the Galaxy. We have searched for spectral lines in the energy range 5--300 GeV using 3.7 years of data, reprocessed with updated instrument calibrations and an improved energy dispersion model compared to the previous Fermi-LAT Collaboration line searches. We searched in five regions selected to optimize sensitivity to different theoretically-motivated dark matter density distributions. We did not find any globally significant lines in our a priori search regions and present 95% confidence limits for annihilation cross sections of self-conjugate WIMPs and decay lifetimes. Our most significant fit occurred at 133 GeV in our smallest search region and had a local significance of 3.3 standard deviations, which translates to a global significance of 1.5 standard deviations. We discuss potential systematic effects in this search, and examine the feature at 133 GeV in detail. We find that both the use of reprocessed data and of additional information in the energy dispersion model contribute to the reduction in significance of the line-like feature near 130 GeV relative to significances reported in other works. We also find that the feature is narrower than the LAT energy resolution at the level of 2 to 3 standard deviations, which somewhat disfavors the interpretation of the 133 GeV feature as a real WIMP signal.