I found the introduction of this article of A. Furman (mainly quoting results of Furstenberg) quite useful in summarizing the big picture:

http://homepages.math.uic.edu/~furman/preprints/fb.pdf

In particular, you can define the universal boundary (boundary = compact Hausdorff $G$-space such that the action is minimal and strongly proximal) for any locally compact group $G$, and $G$ is amenable if and only if the universal boundary is trivial. It looks like this universal boundary is what is usually meant by the Poisson-Furstenberg boundary of a locally compact group.

As far as I know, if the group has exponential growth, that in itself doesn't tell you anything about the boundary.

Edit: I found the introduction of this article of A. Furman (mainly quoting results of Furstenberg) useful for one approach to non-amenability, although it maybe isn't quite what you are looking for:

http://homepages.math.uic.edu/~furman/preprints/fb.pdf

In particular, you can define the universal boundary (boundary = compact Hausdorff $G$-space such that the action is minimal and strongly proximal) for any locally compact group $G$, and $G$ is amenable if and only if the universal boundary is trivial. It looks like this universal boundary $B(G)$ is what is meant by the Furstenberg boundary of a locally compact group.

Given a random walk on $G$ that generates the group, one can then define its Poisson(-Furstenberg) boundary. Depending on the measure, this can be larger than $B(G)$. If I am reading Furstenberg's paper correctly, for a semisimple Lie group they can all be realized as covers of $B(G)$, so $B(G)$ is the smallest Poisson boundary; not sure how far this generalizes.