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Employing new infrared radial velocities, we have computed orbits of the cool giants in three southern S-type symbiotic systems. The orbit for V1044 Cen, an M5.5 giant, has a period of 985 days and a modest eccentricity of 0.16. Hen 3-1213 is a K4 giant, yellow symbiotic with an orbital period of 533 days and a similar eccentricity of 0.18. For the M2 giant SS 73-96 the orbital period is 828 days, and this system has a somewhat larger eccentricity of 0.26. Measurement of the H i Paschen δ emission lines, which may at least partially reflect the motion of the secondary in SS 73-96, results in a mass ratio of 2.4 for the M giant relative to the presumed white dwarf. The estimated orbital inclinations of V1044 Cen and Hen 3-1213 are low, about 40°. However, for SS 73-96 the predicted inclination is 90°, and so an ephemeris for eclipses of the secondary or the hot nebula surrounding it is provided. A search of the orbital velocity residuals of V1044 Cen and SS 73-96 for pulsation periods produced no realistic or convincing period for either star.