I have had best luck with finazzle. Next best is to steam clean the grout. I never tried bleach but I guess it would work on bright white only.

I have used comet/water paste and soft scrub with bleach on a light colored grout (morning fog-a very light grey) for seven years with good results. I have even used clorox cleanup and then scrubbed the grout with a nail brush.
It does not seem to change the color of the grout at all.
When I had tile installed in a very large kitchen/breakfast nook the installer moaned at my grout choice because it was so light. He said I would never be able to keep it clean. He told me to use nothing to clean the floor but warm water and told me that soft scrub and a nail brush worked well on the grout. He was right. I had previously used Comet paste on the foyer grout (medium light cream color) and used it on two of the three bathrooms with light colored grout so I had no fear in using it in the kitchen.
Try an inconspicuous spot where if something adverse happens you will be the only one to notice.

The pros say that bleach is a no-no on grout. I forgot why, though. Now Oxygen bleach is okay. If the Ox Bleach doesn't get it white enough, you can try a grout colorant.

I have used Soft Scrub with bleach on our light colored grout (entrance floor) with a nail brush and had excellent results. Make sure you rinse well. Apply a good sealant with a tiny paint brush and you are good for years!