There has not yet been a single study of any kind of rhythmicity in Archaea. Most of those microorganisms live in strange places - miles deep under the surface of the earth, in rocks, in ice, on the ocean floors and in the hydrothermal vents. They mostly do not inhabit rhythmic environments, so perhaps they do not need to have clocks - but it would be really nice to know if that is really the case:
Old Faithful, the famous geyser in Yellowstone park contains Archea. As the geyser erupts every 45 minutes or so, the microbes are suddenly exposed to very different environment: light, turbulence, lower temperature. Should we expect them to evolve a 45-minute clock that will help them predict the eruption so they can limit some sensitive biochemical reactions to the quiet periods and switch on the defenses agains light and cold every 45 minutes?



For instance in a rodent called Octogon degus, olfactory signals from one individual to another mutually entrain their rhtyhms (I think it has to be one male and one female, though). Other rodents may synchronize their rhtyhms when kept in group conditions without any rhyhtmic clues from the environment.

There is also a model (I do not know how well it is empirically supported) in which there is a feedback between the clock of dinoflagellate Gonyolax polyedra (I think it has been recently reclassified and renamed, but all teh literature used this name) and its environment, i.e., the little protists affect their environment which in turn entrains their clock which in turn makes teh environment rhythmmic (salinity, or pH or something), and so on.

Those are just some examples at the time-scales of day-to-day activity of organisms. At an evolutionary scale, many organisms evolve the rhythms in ways that synchronize with other species (usually trying to avoid each other). There are a number of cases of evolutionary arms-races around the circadian clock, as well as cases of competitive exlusion in time (as opposed to space). Some golden spiny mice in Israel - two closely related species - are sympatric (live in the same place) in one canyon. One of the species remains nocturnal, while the other one is forced to turn diurnal (only in that geographical area), with con-commitant changes in the anatomy and physiology of the eyes, etc.