We report the initial results from a Spitzer GO-1 program to obtain low-resolution, mid-infrared spectra of infrared luminous galaxies at z ~ 1-2. This paper presents the spectra of eight sources observed with the Spitzer Infrared Spectrograph (IRS). Of the eight spectra, six have mid-IR spectral features, either emission from polycyclic aromatic hydrocarbon (PAH) or silicate absorption. Based on these mid-IR features, the inferred six redshifts are in the range of 1.8-2.6. The remaining two spectra detect only strong continua and thus do not yield redshift information. Strong, multiple PAH emission features are detected in two sources, and weak PAH emission is detected in another two. These data provide direct evidence that PAH molecules are present and directly observable in ultraluminous infrared galaxies (ULIRGs) at z ~ 2. The six sources with measured redshifts are dusty, infrared luminous galaxies at z ~ 2 with estimated L_bol ~ 10¹³ L. Of the eight sources, two appear to be starburst dominated; two with only power-law continua are probably type 1 QSOs; and the remaining four are likely composite systems containing a buried active galactic nucleus (AGN) and a starburst component. Since half of our sample contains optically faint sources with R ≥ 25.5 mag (Vega), our results demonstrate the potential of using mid-infrared spectroscopy, especially the aromatic and silicate features produced by small dust grains, to directly probe optically faint and infrared luminous populations at high redshift.