Spectroscopic orbits have been reported for six unseen companions orbiting solar-type stars with minimum possible masses in the range 0.5-10 Jupiter masses. The four least massive companions, around 51 Peg, 47 UMa, 55 Cnc, and τ Boo, have nearly circular orbits, while the two most massive companions, around HD 114762 and 70 Vir, have eccentricities of 0.35 and 0.40. We compare the orbital eccentricities of these six planet candidates with the eccentricities of the planets in the solar system, of the three planets found around the pulsar PSR B1957+12, and of the low-mass secondaries in a subsample of the spectroscopic binaries from the Carney-Latham proper-motion survey. The distribution of eccentricities for the combined samples displays a striking pattern: the companions with masses smaller than about 5 Jupiter masses have circular orbits, while the more massive companions have eccentric orbits. We outline four possible scenarios that might have produced this pattern of eccentricity versus mass.