The formation of columnar eddies in a rapidly rotating environment is often attributed to nonlinear processes, acting on the nonlinear time scale $l/|\bm{u}|$. We argue that this is not the whole story, and that linear wave propagation can play an important role, at least on the short time scale of $\Omega^{-1}$. In particular, we consider the initial value problem of a compact blob of vorticity (an eddy) sitting in a rapidly rotating environment. We show that, although the energy of the eddy disperses in all directions through inertial wave propagation, the axial components of its linear impulse and angular momentum disperse along the rotation axis only, remaining confined to the cylinder which circumscribes the initial vortex blob. This confinement has a crucial influence on the manner in which energy disperses from the eddy, with the energy density within the tangent cylinder remaining much higher than that outsid (i.e. decaying as $t^{-1}$ inside the cylinder and $t^{-3/2}$ outside). When the initial conditions consist of an array of vortex blobs the situation is more complicated, because the energy density within the tangent cylinder of any one blob is eventually swamped by the radiation released from all the other blobs. Nevertheless, we would expect that a turbulent flow which starts as a collection of blobs of vorticity will, for times of order $\Omega^{-1}$, exhibit columnar vortices, albeit immersed in a random field of inertial waves. Laboratory experiments are described which do indeed show the emergence of columnar eddies through linear mechanisms, though these experiments are restricted to the case of inhomogeneous turbulence. Since the Rossby number in the experiments is of the order of unity, this suggests that linear effects can still influence and shape turbulence when nonlinear processes are also operating.