We consider the problem of triangulating a d-dimensional uniform grid of d-cubes that is cut by a k-dimensional affine subspace. The goal is to obtain a triangulation with bounded aspect ratio. To achieve this goal, we allow some of the box faces near the affine subspace to be displaced. This problem has applications to finite element mesh generation. For general d and k, the bound on aspect ratio that we attain is double-exponential in d. For the important special case of d = 3, the aspect ratio bound is small enough that the technique is useful in practice.