We present optical and near-infrared (NIR) photometry for three gamma-ray burst supernovae (GRB-SNe): GRB 120729A, GRB 130215A/SN 2013ez, and GRB 130831A/SN 2013fu. For GRB 130215A/SN 2013ez, we also present optical spectroscopy at t-t(0) = 16.1 d, which covers rest-frame 3000-6250 angstrom. Based on Fell lambda 5169 and Sill lambda 6355, our spectrum indicates an unusually low expansion velocity of similar to 4000-6350 km s(-1), the lowest ever measured for a GRB-SN. Additionally, we determined the brightness and shape of each accompanying SN relative to a template supernova (SN 1998bw), which were used to estimate the amount of nickel produced via nucleosynthesis during each explosion. We find that our derived nickel masses are typical of other GRB-SNe, and greater than those of SNe Ibc that are not associated with GRBs. For GRB 130831A/SN 2013fu, we used our well-sampled R-band light curve (LC) to estimate the amount of ejecta mass and the kinetic energy of the SN, finding that these too are similar to other GRB-SNe. For GRB 130215A, we took advantage of contemporaneous optical/NIR observations to construct an optical/NIR bolometric LC of the afterglow. We fit the bolometric LC with the millisecond magnetar model of Zhang & Meszros (2001, ApJ, 552, L35), which considers dipole radiation as a source of energy injection to the forward shock powering the optical/NIR afterglow. Using this model we derive an initial spin period of P = 12 ms and a magnetic field of B = 1.1 x 10(15) G, which are commensurate with those found for proposed magnetar central engines of other long-duration GRBs.