Dynamic simulations based on a slender-body treatment of fibers coupled with a pseudo-spectral solution of the Navier-Stokes equations are used to explore the settling behavior of dilute suspensions of fibers subject to periodic boundary conditions. The most striking result of fluid inertia on the settling of a single fiber is to cause its rotation toward an orientation with its long axis perpendicular to gravity. However, this rotation does not prevent an instability of a homogeneous suspension of finite Reynolds number fibers to particle concentration variations similar to that observed in suspensions of fibers in Stokes flow. While the instability in Stokes flow tends to produce a single permanent streamer of fibers within a periodic cell, finite-Reynolds-number settling fibers exhibit a dynamic structure in which packets of fibers form and disperse continually. The inertial rotation of fibers toward horizontal alignment competes with the rotation of fibers toward vertical alignment caused by the flow driven by fiber concentration variations. The net result is that fibers tend to be more horizontally aligned at higher Reynolds numbers and lower particle concentrations. The mean settling velocity in the inertial settling suspension can exceed the settling velocity of an isolated vertically oriented fiber, but it reaches a statistically steady state and is more modest than the velocities observed in simulations of Stokesian sedimenting fibers in periodic cells.