  We present a spectroscopic survey using the MMT/Hectospec fiber spectrograph
of 24 micron sources selected with the Spitzer Space Telescope in the Spitzer
First Look Survey. We report 1296 new redshifts for 24 micron sources,
including 599 with f(24micron) > 1 mJy. Combined with 291 additional redshifts
for sources from the Sloan Digital Sky Survey (SDSS), our observing program was
highly efficient and is ~90% complete for i' < 21 mag and f(24micron) > 1 mJy,
and is 35% complete for i' < 20.5 mag and 0.3 mJy < f(24micron) < 1 mJy. Our
Hectospec survey includes 1078 and 168 objects spectroscopically classified as
galaxies and QSOs, respectively. Combining the Hectospec and SDSS samples, we
find 24 micron-selected galaxies to z < 0.98 and QSOs to z < 3.6, with mean
redshifts of <z(gal)> = 0.27 and <z(QSO)> =1.1. As part of this publication, we
include the redshift catalogs and the reduced spectra; these are also available
online (http://mips.as.arizona.edu/~papovich/fls) and through the NASA/IPAC
Infrared Science Archive (http://irsa.ipac.caltech.edu).
