We study random walks in a random environment on the d-dimensional lattice Z^d with a uniform local drift. The environment is described as a stationary field of random vectors satistying the, so called, non-nestlig condition. The random vectors at different sites may depend on each other, but their dependence range is finite. We prove the LLN for trajectories of the walk under the probability obtained by averaging the laws of the random walks with respect to the environment (the annealed probability). The techniques developped in the discrete case can be applied to prove related results for diffusions with random drift ( dx(t))=u(omega,x(t))dt+dw(t), x(0)=0 ). Moreover, one can show the existence of a probability measure equivalent to the underlying probability (corresponding to the Eulerian velocity field) under which the particle Lagrangian velocity observations are stationary.