We study the vibrational relaxation dynamics and the reorientation dynamics of HDO molecules in binary water-dimethyl sulfoxide (DMSO) and water-acetone mixtures with polarization-resolved femtosecond mid-infrared spectroscopy. For low solute concentrations we observe a slowing down of the reorientation of part of the water molecules that hydrate the hydrophobic methyl groups of DMSO and acetone. For water-DMSO mixtures the fraction of slowed-down water molecules rises much steeper with solute concentration than for water-acetone mixtures, showing that acetone molecules show significant aggregation already at low concentrations. At high solute concentrations, the vibrational and reorientation dynamics of both water-DMSO and water-acetone mixtures show a clear distinction between the dynamics of water molecules donating hydrogen bonds to other water molecules and the dynamics of water donating a hydrogen bond to the S=O/C=O group of the solute. For water-DMSO mixtures both types of water molecules show a very slow reorientation. The water molecules forming hydrogen bonds to the S=O group reorient with a time constant that decreases from 46 +/- 14 ps at X-DMSO = 0.33 to 13 +/- 2 ps at X-DMSO = 0.95. The water molecules forming hydrogen bonds to the C=O group of acetone show a much faster reorientation with a time constant that decreases from 6.1 +/- 0.2 ps at X-acet = 0.3 to 2.96 +/- 0.05 ps at X-acet = 0.9. The large difference in reorientation time constant of the solute-bound water for DMSO and acetone can be explained from the fact that the hydrogen bond between water and the S=O group of DMSO is much stronger than the hydrogen bond between water and the C=O group of acetone. We attribute the strongly different behavior of water in DMSO-rich and acetone-rich mixtures to their difference in molecular shape.