Let G subset of GL(n) be a finite group without pseudo-reflections. We present an algorithm to compute and verify a candidate for the Cox ring of a resolution X -> 4 C-n/G, which is based just on the geometry of the singularity C-n/G, without further knowledge of its resolutions. We explain the use of our implementation of the algorithms in Singular. As an application, we determine the Cox rings of resolutions X -> C-3/G for all G subset of GL(3) with the aforementioned property and of order vertical bar G vertical bar <= 12. We also provide examples in dimension 4. (C) 2016 Elsevier Inc. All rights reserved.