Late-type evolved stars include the variable red giants, like Mira and semi-regular variables, and the pulsating supergiant RV Tauri stars. Despite this, K-type giants in the past were generally believed to be constant in photometry and radial velocity, but in the last decade it has been discovered that this is not necessarily the case. This was the motivation for spectroscopic observations of 44 late-type evolved stars to be carried out over three years, in the hope of helping to determine how common K giants variable in radial velocity are and whether the variations are related to those seen for other late-type evolved stars. The observations were obtained at Mount John University Observatory and used with digital cross-correlation to achieve relative radial velocities of 50 m/s precision. At this precision all but eight of the observed stars were found to be variable in radial velocity. Thirteen stars also had broad-band photometry taken, because any one of orbital motion, rotation with starspots, pulsation and the motion of large convective cells could cause apparent radial-velocity variations. Knowledge of the relationship between light and radial-velocity variations can help distinguish between these different mechanisms. The timescales for these different mechanisms were also estimated to help determine which was responsible for the variations observed. Generally the observed K-giant radial-velocity timescales were of the order of a few hundred days, which indicates they are due to one of binary motion, rotational modulation or non-acoustic non-radial pulsation. However, for M giants and supergiants the situation was very different with both short-term and long-term radial-velocity timescales being found in most stars. In most cases the short-term timescale is due to acoustic pulsation, while the long-term timescale could be due to any one of binary motion, surface features (including the motion of large convective cells) or non-acoustic non-radial pulsation.