I don't think we have such a lower bound: from a “theoretical" point of view (in the sense of the post), your processor could walk on the cube of memory and collect each bit one by one. Each move+read costs O(1) (if you move correctly), so you get O(n) to read the whole cube of n bits.

If you require the full scan to be done in a specific order however, indeed, in the worse case, you have to go from one end of the cube to the other between each reads, which incurs a O(n^{1/3}) multiplicative cost. Note that this does not constitute a theoretical lower-bound: it might be possible to detect those jumps and use the time spent in a corner to store a few values that will become useful later. This does look like a fun computational problem, I don't know it's exact worst-case complexity.