  We examine a class of random walks in random environments on $\mathbb{Z}$
with bounded jumps, a generalization of the classic one-dimensional model. The
environments we study have i.i.d. transition probability vectors drawn from
Dirichlet distributions. For this model, we characterize recurrence and
transience, and in the transient case we characterize ballisticity. For
ballisticity, we give two parameters, $\kappa_0$ and $\kappa_1$. The parameter
$\kappa_0$ governs finite trapping effects, and $\kappa_1$ governs repeated
traversals of arbitrarily large regions of the graph. We show that the walk is
right-transient if and only if $\kappa_1>0$, and in that case it is ballistic
if and only if $\min(\kappa_0,\kappa_1)>1$.
