Every such graph is generically globally rigid . in $E^2$. A 3-connected chordal graph can be built by starting with a triangle and then sequentially attaching new vertices to at least 3 previous ones. See this paper for some explicit statements. This idea generalizes to any dimension.
In fact, any generic (or even general position) framework for such a graph will be universally rigid , in $E^2$, ie. it has no equivalent and non-congruent frameworks in any dimension. Such a graph is called generically universally rigid . in $E^2$. (Note that you need to be a bit careful with universal rigidity as there are graphs that have some generic frameworks that are universally rigid , in $E^2$, and other generic frameworks that are not universally rigid.)rigid in $E^2$.)
