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Radial velocities from the 2.1 m telescope at McDonald Observatory supplemented with radial velocities from the coudé feed telescope at Kitt Peak National Observatory provide new precise orbits for the double-lined spectroscopic binaries RR Lyn (A3/A8/A6), 12 Boo (F8 IV), and HR 6169 (A2 V). We derive orbital dimensions (a1 sin i and a2 sin i) and minimum masses (m1 i and m2 i) with accuracies of 0.06%-0.9%. The three systems, which have V magnitudes of 5.53, 4.82, and 6.42, respectively, are all sufficiently bright that they are easily within the grasp of modern optical interferometers and so afford the prospect, when our spectroscopic observations are complemented by interferometric observations, of fully determined orbits, precise masses, and distances. In the case of RR Lyn, which is also a detached eclipsing binary with a well-determined orbital inclination (i = 87fdg45 ± 0fdg11), we are able to determine the semimajor axis of the relative orbit, a = 29.32 ± 0.04 R⊙; primary and secondary radii of 2.57 ± 0.02 and 1.59 ± 0.03 R⊙, respectively; and primary and secondary masses of 1.927 ± 0.008 and 1.507 ± 0.004 M⊙, respectively. Comparison of our new systemic velocity determination, γ = -12.03 ± 0.04 km s-1, with the earlier one of Kondo, γ = -11.61 ± 0.30 km s-1, shows no evidence of any change in the systemic velocity in the 40 yr separating the two measurements, a null result that neither confirms nor contradicts the presence of the low-mass third component proposed by Khaliullin & Khaliullina. Our spectroscopic orbit of 12 Boo is more precise than that of Boden et al. but confirms their results about this system. Our analysis of HR 6169 has produced a major improvement in its orbital elements. The minimum masses of the primary and secondary are 2.20 ± 0.01 and 1.64 ± 0.02 M⊙, respectively. Although all three systems have eccentric orbits, the six components of the systems are either pseudosynchronously rotating or very nearly so.