  We prove lower bounds on the localization length of eigenfunctions in the
three-dimensional Anderson model at weak disorders. Our results are similar to
those obtained by Schlag, Shubin and Wolff for dimensions one and two. We prove
that with probability one, most eigenfunctions have localization lengths
bounded from below by $O(\frac{\lambda^{-2}}{\log\frac1\lambda})$, where
$\lambda$ is the disorder strength. This is achieved by time-dependent methods
which generalize those developed by Erd\"os and Yau to the lattice and
non-Gaussian case. In addition, we show that the macroscopic limit of the
corresponding lattice random Schr\"odinger dynamics is governed by the linear
Boltzmann equations.
