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Dusty star-forming galaxies are among the most prodigious systems at highredshift (z>1), characterized by high star formation rates and huge dustreservoirs. The bright end of this population has been well characterized inrecent years, but considerable uncertainties remain for fainter dustystar-forming galaxies, which are responsible for the bulk of star formation athigh 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 highredshift faint dusty galaxies using millimeter observations with ALMA. Weobtained ALMA 1.1 mm mosaic images for three strong-lensing galaxy clustersfrom the Frontier Fields survey. The 2'x2' mosaics overlap with the deep HSTWFC3/IR footprints and encompass the high magnification regions of eachcluster. The combination of extremely high ALMA sensitivity and themagnification power of these clusters allows us to systematically probe thesub-mJy population of dusty star-forming galaxies over a large surveyed area.  We present a description of the reduction and analysis of the ALMA continuumobservations 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 rmssensitivities of 55, 59 and 71 $\mu$Jy beam$^{-1}$ respectively. We detect 12dusty star-forming galaxies at S/N>5.0 across the three clusters, all of thempresenting coincidence with NIR detected counterparts in the HST images. Noneof the sources fall close to the lensing caustics, thus they are not stronglylensed. The observed 1.1 mm flux densities for the total sample of galaxiesrange from 0.41 to 2.82 mJy, with observed effective radii spanning <0.05" to0.37"$\pm$0.21". The lensing-corrected sizes of the detected sources appear tobe in the same range as those measured in brighter samples, albeit withpossibly larger dispersion.

The ALMA Frontier Fields survey I: 1.1 mm continuum detections in Abell 2744, MACSJ0416.1-2403 and MACSJ1149.5+2223