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Few galaxies have been found between the redshift ranges z < ~1 probed bymagnitude-limited surveys and z > ~3 probed by Lyman-break surveys. Comparisonof galaxy samples at lower and higher redshift suggests that large numbers ofstars were born and the Hubble sequence began to take shape at the intermediateredshifts 1<z<3, but observational challenges have prevented us from observingthe process in much detail. We present simple and efficient strategies that canbe used to find large numbers of galaxies throughout this important butunexplored redshift range. All the strategies are based on selecting galaxiesfor spectroscopy on the basis of their colors in ground-based images takenthrough a small number of optical filters: GRi for redshifts 0.85<z<1.15, GRzfor 1<z<1.5, and UGR for 1.4<z<2.1 and 1.9<z<2.7. The performance of ourstrategies is quantified empirically through spectroscopy of more than 2000galaxies at 1<z<3.5. We estimate that more than half of the UV-luminositydensity at 1<z<3 is produced by galaxies that satisfy our color-selectioncriteria. Our methodology is described in detail, allowing readers to deviseanalogous selection criteria for other optical filter sets.Comment: 13 pages including 20 figures. Accepted for publication in the Ap

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