Most diy stores sell white stuff not unlike Tipp-Ex but more durable. It comes in a pot with a small brush or in a felt tip pen type thing. Try to get the pen thing, it costs more but is easier and you waste less.It covers the grout with a fresh coat of fairly durable white stuff. The effect is not as good as re-grouting but way easier, even allowing for the fact that the grout has to be grease, soap and limescale free for the stuff to adhere.
I use some tile groutinmg whitener which I buy from our Betterware catalogue. paint it on (takes ages) leave it for two hours then use a scourer to go over any bits on the tiles. It looks 100% better when I've done it and lasts about 2 months before they look grubbyish again. Pain in the neck to do but worth it.
Thanks for trying Hails, I do appreciate the effort. From what you say, it takes a long time, a lot of work and after two months it needs doing again. Regrouting could be the best option for long term looks(and expensive, I know).