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Using a systematic broadband search technique, we have carried out a survey for large Ly alpha nebulae (or Ly alpha "blobs") at 2 less than or similar to z less than or similar to 3 within 8.5 deg(2) of the NOAO Deep Wide-Field Survey Bootes field, corresponding to a total survey comoving volume of approximate to 10(8) h(70)(-3) Mpc(3). Here, we present our spectroscopic observations of candidate giant Ly alpha nebulae. Of 26 candidates targeted, 5 were confirmed to have Ly alpha emission at 1.7 less than or similar to z less than or similar to 2.7, 4 of which were new discoveries. The confirmed Ly alpha nebulae span a range of Ly alpha equivalent widths, colors, sizes, and line ratios, and most show spatially extended continuum emission. The remaining candidates did not reveal any strong emission lines, but instead exhibit featureless, diffuse, blue continuum spectra. Their nature remains mysterious, but we speculate that some of these might be Ly alpha nebulae lying within the redshift desert (i.e., 1.2 less than or similar to z less than or similar to 1.6). Our spectroscopic follow-up confirms the power of using deep broadband imaging to search for the bright end of the Ly alpha nebula population across enormous comoving volumes.