Cross-correlations of galaxy positions and galaxy shears with maps of gravitational lensing of the cosmic microwave background (CMB) are sensitive to the distribution of large-scale structure in the Universe. Such cross-correlations are also expected to be immune to some of the systematic effects that complicate correlation measurements internal to galaxy surveys. We present measurements and modeling of the cross-correlations between galaxy positions and galaxy lensing measured in the first three years of data from the Dark Energy Survey with CMB lensing maps derived from a combination of data from the 2500 deg2 SPT-SZ survey conducted with the South Pole Telescope and full-sky data from the Planck satellite. The CMB lensing maps used in this analysis have been constructed in a way that minimizes biases from the thermal Sunyaev Zel'dovich effect, making them well suited for cross-correlation studies. The total signal-to-noise of the cross-correlation measurements is 23.9 (25.7) when using a choice of angular scales optimized for a linear (nonlinear) galaxy bias model. We use the cross-correlation measurements to obtain constraints on cosmological parameters. For our fiducial galaxy sample, which consist of four bins of magnitude-selected galaxies, we find constraints of ωm=0.272-0.052+0.032 and S8σ8ωm/0.3=0.736-0.028+0.032 (ωm=0.245-0.044+0.026 and S8=0.734-0.028+0.035) when assuming linear (nonlinear) galaxy bias in our modeling. Considering only the cross-correlation of galaxy shear with CMB lensing, we find ωm=0.270-0.061+0.043 and S8=0.740-0.029+0.034. Our constraints on S8 are consistent with recent cosmic shear measurements, but lower than the values preferred by primary CMB measurements from Planck.