Indeed, the answer is 2+ϵ in most senses. For example, consider a wedge in Z3 which is all the points (x,y,z) such that |z|<f(x) for some specific increasing function f:N->N .
These wedges are transient already for pretty slowly growing functions. If you take f(x)=xϵ which correspond, in the sense of volume growth, to 2+ϵ dimensions, then it is transient for any ϵ>0.
One can get even more refined results by taking f(x)=log(x)a, in which case the wedge is transient if and only if a>1.
You can find more information in the book of Lyons with Peres.