Temperature control algorithms in molecular dynamics (MD) simulations are necessary to study isothermal systems. However, these thermostatting algorithms alter the velocities of the particles and thus modify the dynamics of the system with respect to the microcanonical ensemble, which could potentially lead to thermostat-dependent dynamical artifacts. In this study, we investigate how six well-established thermostat algorithms applied with different coupling strengths and to different degrees of freedom affect the dynamics of various molecular systems. We consider dynamic processes occurring on different times scales by measuring translational and rotational self-diffusion as well as the shear viscosity of water, diffusion of a small molecule solvated in water, and diffusion and the dynamic structure factor of a polymer chain in water. All of these properties are significantly dampened by thermostat algorithms which randomize particle velocities, such as the Andersen thermostat and Langevin dynamics, when strong coupling is used. For the solvated small molecule and polymer, these dampening effects are reduced somewhat if the thermostats are applied to the solvent alone, such that the solute's temperature is maintained only through thermal contact with solvent particles. Algorithms which operate by scaling the velocities, such as the Berendsen thermostat, the stochastic velocity rescaling approach of Bussi and co-workers, and the Nosé-Hoover thermostat, yield transport properties that are statistically indistinguishable from those of the microcanonical ensemble, provided they are applied globally, i.e. coupled to the system's kinetic energy. When coupled to local kinetic energies, a velocity scaling thermostat can have dampening effects comparable to a velocity randomizing method, as we observe when a massive Nose-Hoover coupling scheme is used to simulate water. Correct dynamical properties, at least those studied in this paper, are obtained with the Berendsen thermostat applied globally, despite the fact that it yields the wrong kinetic energy distribution.