We use data from the first 100 deg ^ { 2 } field observed by the South Pole Telescope ( SPT ) in 2008 to measure the angular power spectrum of temperature anisotropies contributed by the background of dusty star-forming galaxies ( DSFGs ) at millimeter wavelengths . From the auto and cross-correlation of 150 and 220 GHz SPT maps , we significantly detect both Poisson distributed and , for the first time at millimeter wavelengths , clustered components of power from a background of DSFGs . The spectral indices of the Poisson and clustered components are found to be \bar { \alpha } _ { 150 - 220 } ^ { P } = 3.86 \pm 0.23 and \alpha _ { 150 - 220 } ^ { C } = 3.8 \pm 1.3 , implying a steep scaling of the dust emissivity index \beta \sim 2 . The Poisson and clustered power detected in SPT , BLAST ( at 600 , 860 , and 1200 GHz ) , and Spitzer ( 1900 GHz ) data can be understood in the context of a simple model in which all galaxies have the same graybody spectrum with dust emissivity index of \beta = 2 and dust temperature T _ { d } = 34 K. In this model , half of the 150 GHz background light comes from redshifts greater than 3.2 . We also use the SPT data to place an upper limit on the amplitude of the kinetic Sunyaev–Zel ’ dovich power spectrum at \ell = 3000 of 13 \mu { K } ^ { 2 } at 95 % confidence .