  Dusty star-forming galaxies are among the most prodigious systems at high
redshift (z>1), characterized by high star formation rates and huge dust
reservoirs. The bright end of this population has been well characterized in
recent years, but considerable uncertainties remain for fainter dusty
star-forming galaxies, which are responsible for the bulk of star formation at
high redshift and thus play a key role in galaxy growth and evolution.
  In this first paper of our series, we describe our methods for finding high
redshift faint dusty galaxies using millimeter observations with ALMA. We
obtained ALMA 1.1 mm mosaic images for three strong-lensing galaxy clusters
from the Frontier Fields survey. The 2'x2' mosaics overlap with the deep HST
WFC3/IR footprints and encompass the high magnification regions of each
cluster. The combination of extremely high ALMA sensitivity and the
magnification power of these clusters allows us to systematically probe the
sub-mJy population of dusty star-forming galaxies over a large surveyed area.
  We present a description of the reduction and analysis of the ALMA continuum
observations for the galaxy clusters Abell 2744 (z=0.308), MACSJ0416.1-2403
(z=0.396) and MACSJ1149.5+2223 (z=0.543), for which we reach observed rms
sensitivities of 55, 59 and 71 $\mu$Jy beam$^{-1}$ respectively. We detect 12
dusty star-forming galaxies at S/N>5.0 across the three clusters, all of them
presenting coincidence with NIR detected counterparts in the HST images. None
of the sources fall close to the lensing caustics, thus they are not strongly
lensed. The observed 1.1 mm flux densities for the total sample of galaxies
range from 0.41 to 2.82 mJy, with observed effective radii spanning <0.05" to
0.37"$\pm$0.21". The lensing-corrected sizes of the detected sources appear to
be in the same range as those measured in brighter samples, albeit with
possibly larger dispersion.
