Whitening toothpaste can also be used to clean grout in a relatively mild manner.
Vinegar can be used to scrub grout, as can a mix of bicarbonate of soda and vinegar.
For serious grout stains, reach for the bleach.
Dip and brush into a small amount of bleach and scrub directly onto stains.
Or, make a paste with bicarbonate of soda and bleach.
Leave this on the stain for 15 minutes and then scrub it away before rinsing.