We had a similar problem in our bathroom. Tried every cleaner product available at HD, used a high-powered steam cleaner, bleach, etc.. to no avail. Very frustrating. Looks like maybe the grout wasnt sealed and dirt got into it and there it will forever stay.
We ended up bleaching it, drying it well, and then painting it with grout paint. We went a darker color so a single pass covered it, and we are very happy with the results. It has been about two years and it is lightening a bit, (probably due to lack of maintenance on my end, haven't resealed it) so I plan to repaint it this week and then seal over the paint.