  Magnetic reconnection is invoked as one of the primary mechanisms to produce
energetic particles. We employ large-scale three-dimensional (3D)
particle-in-cell simulations of reconnection in magnetically-dominated
($\sigma=10$) pair plasmas to study the energization physics of high-energy
particles. We identify a novel acceleration mechanism that only operates in 3D.
For weak guide fields, 3D plasmoids / flux ropes extend along the $z$ direction
of the electric current for a length comparable to their cross-sectional
radius. Unlike in 2D simulations, where particles are buried in plasmoids, in
3D we find that a fraction of particles with $\gamma\gtrsim 3\sigma$ can escape
from plasmoids by moving along $z$, and so they can experience the large-scale
fields in the upstream region. These "free" particles preferentially move in
$z$ along Speiser-like orbits sampling both sides of the layer, and are
accelerated linearly in time -- their Lorentz factor scales as $\gamma\propto
t$, in contrast to $\gamma\propto \sqrt{t}$ in 2D. The energy gain rate
approaches $\sim eE_{\rm rec}c$, where $E_{\rm rec}\simeq 0.1 B_0$ is the
reconnection electric field and $B_0$ the upstream magnetic field. The spectrum
of free particles is hard, $dN_{\rm free}/d\gamma\propto \gamma^{-1.5}$,
contains $\sim 20\%$ of the dissipated magnetic energy independently of domain
size, and extends up to a cutoff energy scaling linearly with box size. Our
results demonstrate that relativistic reconnection in GRB and AGN jets may be a
promising mechanism for generating ultra-high-energy cosmic rays.
