We consider a family of spatially non-homogeneous random walks in R^d, d geq 2, with zero drift, which in any dimension can be recurrent or transient depending on the parameters of the walk. This contrasts with the homogeneous case (for example, the symmetric simple random walk in Z^d) where recurrence/transience is determined by the dimension d. We give a description of the diffusion limits of these walks, which in the transient case resembles the skew-product decomposition of Brownian motion. In the recurrent case, the description is given via excursions built from a Bessel process, where each excursion has a similar skew-product decomposition.