The diagonal dimension of a C*-subalgebra was introduced by Li, Liao, and Winter, and is a version of nuclear dimension for diagonal C*-subalgebras. In the case of crossed product C*-algebras, it extracts information on the underlying dynamical systems. Since diagonal dimension restricts the study to nuclear algebras, we propose a generalisation of the dimension that also treats non-nuclear algebras. Our motivation comes from Roe algebras, which are C*-algebras that bridge the study of coarse geometry and operator algebras. The generalised diagonal dimension can be used to recover the asymptotic dimension of a metric space from its (non-uniform) Roe algebra.