We report Doppler measurements for six nearby G- and K-type main-sequence stars that show multiple low-mass companions, at least one of which has planetary mass. One system has three planets, the fourth triple-planet system known around a normal star, and another has an extremely low minimum mass of 18 M_⊕. HD 128311 (K0 V) has two planets (one previously known) with minimum masses (M sin i) of 2.18M_J and 3.21M_J and orbital periods of 1.26 and 2.54 yr, suggesting a possible 2 : 1 resonance. For HD 108874 (G5 V), the velocities reveal two planets (one previously known) having minimum masses and periods of (M sin i_b = 1.36M_J, P_b = 1.08 yr) and (M sin i_c = 1.02M_J, P_c = 4.4 yr). HD 50499 (G1 V) has a planet with P = 6.8 yr and M sin i = 1.7M_J, and the velocity residuals exhibit a trend of -4.8 m s^-1 yr^-1, indicating a more distant companion with P > 10 yr and minimum mass of 2M_J. HD 37124 (G4 IV-V) has three planets, one having M sin i = 0.61M_J and P = 154.5 days, as previously known. We find two plausible triple-planet models that fit the data, both having a second planet near P = 840 days, with the more likely model having its third planet in a 6 yr orbit and the other one in a 29 day orbit. For HD 190360, we confirm the planet having P = 7.9 yr and M sin i = 1.5M_J as found by the Geneva team, but we find a distinctly noncircular orbit with e = 0.36 ± 0.03, rendering this not an analog of Jupiter as had been reported. Our velocities also reveal a second planet with P = 17.1 days and M sin i = 18.1 M_⊕. HD 217107 (G8 IV) has a previously known "hot Jupiter" with M sin i = 1.4M_J and P = 7.13 days, and we confirm its high eccentricity, e = 0.13. The velocity residuals reveal an outer companion in an eccentric orbit, having minimum mass of M sin i > 2M_J, eccentricity e ~ 0.5, and a period P > 8 yr, implying a semimajor axis a > 4 AU and providing an opportunity for direct detection. We have obtained high-precision photometry of five of the six planetary host stars with three of the automated telescopes at Fairborn Observatory. We can rule out significant brightness variations in phase with the radial velocities in most cases, thus supporting planetary reflex motion as the cause of the velocity variations. Transits are ruled out to very shallow limits for HD 217107 and are also shown to be unlikely for the prospective inner planets of the HD 37124 and HD 108874 systems. HD 128311 is photometrically variable with an amplitude of 0.03 mag and a period of 11.53 days, which is much shorter than the orbital periods of its two planetary companions. This rotation period explains the origin of periodic velocity residuals to the two-planet model of this star. All of the planetary systems here would be further constrained with astrometry by the Space Interferometry Mission.