In their paper "Existence of positive harmonic functions on groups and on covering manifolds", Bougerol & Elie give an overview of the connection between the three properties growth, amenability, and existence of bounded/positive harmonic functions.

In short, under the right conditions, amenability is equivalent to the existence of (non-trivial) bounded harmonic functions, polynomial growth implies no positive harmonic functions, and exponential growth implies existence of positive (non-trivial) harmonic function.

More accurately:

[Azencott] A non-amenable compactly generated locally compact group equipped with an adapted probability measure admit non-constant bounded (hence also positive) harmonic function.

[Guivarc'h, Kaimanovich, Alexopoulos] A connected amenable compactly generated locally compact group equipped with an adapted, centered, with a compactly supported continuous density probability measure, admits no bounded harmonic functions other than the constants.

[Hebish & Saloff Coste] A compactly generated locally compact group with polynomial growth equipped with an adapted, symmetric, with a compactly supported continuous density probability measure, admits no positive harmonic functions other than the (positive) constants.

Bougerol & Elie then prove:

- A compactly generated locally compact group that admits a continuous homomorphism into a almost connected group and such that the closure of the image has exponential growth, equipped with an adapted, centered, with a continuous density probability measure that has a third moment, admits a non-constant positive harmonic function.

Most of the references are in French, so no wonder I didn't find them before. :)

I would appreciate any comments on this, and I thank all the respondents!