We study the complexity of approximate counting Constraint Satisfaction Problems (#CSPs) in a bounded degree setting. Specifically, given a Boolean constraint language $Γ$ and a degree bound $Δ$, we study the complexity of #CSP$_Δ(Γ)$, which is the problem of counting satisfying assignments to CSP instances with constraints from $Γ$ and whose variables can appear at most $Δ$ times. Our main result shows that: (i) if every function in $Γ$ is affine, then #CSP$_Δ(Γ)$ is in FP for all $Δ$, (ii) otherwise, if every function in $Γ$ is in a class called IM$_2$, then for all sufficiently large $Δ$, #CSP$_Δ(Γ)$ is equivalent under approximation-preserving (AP) reductions to the counting problem #BIS (the problem of counting independent sets in bipartite graphs) (iii) otherwise, for all sufficiently large $Δ$, it is NP-hard to approximate the number of satisfying assignments of an instance of #CSP$_Δ(Γ)$, even within an exponential factor. Our result extends previous results, which apply only in the so-called "conservative" case.