The Searchlight Scheduling Problem was first studied in 2D polygons, where
the goal is for point guards in fixed positions to rotate searchlights to catch
an evasive intruder. Here the problem is extended to 3D polyhedra, with the
guards now boundary segments who rotate planes of illumination. After carefully
detailing the 3D model, several results are established. The first is a nearly
direct extension of the planar one-way sweep strategy using what we call
exhaustive guards, a generalization that succeeds despite there being no
well-defined notion in 3D of planar "clockwise rotation". Next follow two
results: every polyhedron with r>0 reflex edges can be searched by at most r^2
suitably placed guards, whereas just r guards suffice if the polyhedron is
orthogonal. (Minimizing the number of guards to search a given polyhedron is
easily seen to be NP-hard.) Finally we show that deciding whether a given set
of guards have a successful search schedule is NP-hard. A number of peripheral
results are proved en route to these central theorems, and several open
problems remain for future work.