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We present a modification to the iodine cell Doppler technique that eliminates the need for an observed stellar template spectrum. For a given target star, we iterate toward a synthetic template spectrum beginning with an existing template of a similar star. We then perturb the shape of this first-guess template to match the program observation of the target star taken through an iodine cell. The elimination of a separate template observation saves valuable telescope time, a feature that is ideally suited for the quick-look strategy employed by the "Next 2000 Stars" (N2K) planet search program. Tests using Keck HIRES (High Resolution Echelle Spectrometer) spectra indicate that synthetic templates yield a short-term precision of 3 m s-1 and a long-term, run-to-run precision of 5 m s-1. We used this new Doppler technique to discover three new planets: a 1.50MJ planet in a 2.1375 day orbit around HD 86081; a 0.71MJ planet in circular, 26.73 day orbit around HD 224693; and a Saturn-mass planet in an 18.179 day orbit around HD 33283. The remarkably short period of HD 86081b bridges the gap between the extremely short period planets detected in the Optical Gravitational Lensing Experiment (OGLE) survey and the 16 Doppler-detected hot Jupiters (P < 15 days), which have an orbital period distribution that piles up at about 3 days. We have acquired photometric observations of two of the planetary host stars with the automated photometric telescopes at Fairborn Observatory. HD 86081 and HD 224693 both lack detectable brightness variability on their radial velocity periods, supporting planetary-reflex motion as the cause of the radial velocity variability. HD 86081 shows no evidence of planetary transits in spite of a 17.6% transit probability. We have too few photometric observations to detect or rule out transits for HD 224693.