There are some companies that specialize in doing this, but I just use a small brush and a product called Tile-X. It works real good. Then get some grout sealer and they'll stay clean longer.
What's the best, quickest and easiest way to clean grout from your bathroom tiles?
Once you know the correct method, then grout can be quite easy
to clean off tiles. If you just follow these simple steps, then you
should have your bathroom clean in no time. It works for me every
time!
Firstly, you need to get rid of any dust off
the tiles, as this will stop you getting to the grout itself. If its
floor tiles you're cleaning - then just use a broom and give them a
quick sweep, or if its on the walls then use a dry cloth to remove
the dust.
Next you need to apply your product. This can
either be bleach which is very effective at cleaning grout, but can
only be used on white or light colours, or it can fade the grout.
You can also use special grout cleaning products, although these may
be quite expensive. Lastly you could use the home made combination of
vinegar and baking soda. Make sure the product is applied in all of
the crevices.
Leave the product to soak into the grout,
make sure it is covering the whole surface. About half an hour
should do the trick.
The dirt should now be very easy to lift off.
You will need to scrub it using a brush that has hard bristles such
as a tooth brush.
You might need to repeat the process a few
times if the stains are really stubborn. If not, simply wipe the
area clean with water then leave to dry.
Good luck with your cleaning, I hope this helped!
This video also gives some really useful grout cleaning tips.