We report multiple Doppler measurements of five nearby FGK main-sequence stars and subgiants obtained during the past 4-6 yr at the Keck Observatory. These stars, namely, HD 183263, HD 117207, HD 188015, HD 45350, and HD 99492, all exhibit coherent variations in their Doppler shifts consistent with a planet in Keplerian motion. The five new planets occupy known realms of planetary parameter space, including a wide range of orbital eccentricities, e = 0-0.78, and semimajor axes, 0.1-3.8 AU, that provide further statistical information about the true distributions of various properties of planetary systems. One of the planets, HD 99492b, has a low minimum mass of 0.112M_Jup = 36M_Earth. Four of the five planets orbit beyond 1 AU. We describe two quantitative tests of the false alarm probability for Keplerian interpretations of measured velocities. The more robust of these involves Monte Carlo realizations of scrambled velocities as a proxy for noise. Keplerian orbital fits to that "noise" yield the distribution of χ to compare with χ from the original (unscrambled) velocities. We establish a 1% false alarm probability as the criterion for candidate planets. All five of these planet-bearing stars are metal-rich, with [Fe/H] > +0.27, reinforcing the strong correlation between planet occurrence and metallicity. From the full sample of 1330 stars monitored at Keck, Lick, and the Anglo-Australian Telescope, the shortest orbital period for any planet is 2.64 days, showing that shorter periods occur less frequently than 0.1% in the solar neighborhood. Photometric observations were acquired for four of the five host stars with an automatic telescope at Fairborn Observatory. The lack of brightness variations in phase with the radial velocities supports planetary-reflex motion as the cause of the velocity variations. No transits were observed, but their occurrence is not ruled out by our observations.