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Infrared radial velocities have been used to derive the first well-determined orbital elements for the cool giants of three symbiotic systems, BF Cyg, V1329 Cyg, and V343 Ser=AS 289. Periods found for BF Cyg and V1329 Cyg from the radial velocity data are in good agreement with periods previously determined for their light variations, and the orbits are circular. Masses for the components of BF Cyg and V1329 Cyg were determined by combining our orbital elements for the cool giants with elements for their hot stars. BF Cyg and V1329 Cyg are shown to be detached binary systems. The third system, V343 Ser, has an orbital period of 450.5 days, a value that is typical for symbiotic binaries. However, the system is atypical because of its somewhat eccentric orbit, with e = 0.135 and the possibility that it is a semidetached system. An orbital inclination of 18° is estimated, indicating that V343 Ser does not eclipse.