We make use of the high photometric precision of Kepler to search for periodic modulations among 14 normal (DA-and DB-type, likely non-magnetic) hot white dwarfs (WDs). In five, and possibly up to seven of the WDs, we detect periodic, ∼ 2 hr to 10 d, variations, with semi-amplitudes of 60-2000 ppm, lower than ever seen in WDs. We consider various explanations: WD rotation combined with magnetic cool spots; rotation combined with magnetic dichroism; rotation combined with hot spots from an interstellar-medium accretion flow; transits by size ∼ 50-200 km objects; relativistic beaming due to reflex motion caused by a cool companion WD; or reflection/reradiation of the primary WD light by a brown-dwarf or giant-planet companion, undergoing illumination phases as it orbits the WD. Each mechanism could be behind some of the variable WDs, but could not be responsible for all five to seven variable cases. Alternatively, the periodicity may arise from UV metal-line opacity, associated with accretion of rocky material, a phenomenon seen in ∼ 50% of hot WDs. Nonuniform UV opacity, combined with WD rotation and fluorescent optical re-emission of the absorbed UV energy, could perhaps explain our findings. Even if reflection by a planet is the cause in only a few of the seven cases, it would imply that hot Jupiters are very common around WDs. If some of the rotation-related mechanisms are at work, then normal WDs rotate as slowly as do peculiar WDs, the only kind for which precise rotation measurements have been possible to date.