If you drop a glass of juice or even water, clean it up right away. The longer it sits, the more time it will have to soak into the grout.
The thing that most consumers do not realize is that grout really needs to be sealed every 2 - 3 years. The best thing to do is use a heavy duty grout cleaner first to see if it helps. Sometimes the grout is so dirty that the only option is using a grout colorant. Take a look at these options for your tile: http://www.mytilebacksplash.com/cleaning-and-maintenance.aspx
I just got done cleaning an entry floor, white, and a bathroom floor with medicinal strength peroxide. Haven't been this clean in years. Just squirted it out of the bottle onto the floor, let it sit a short time and mopped away. and I mean AWAY.. Good kuck.
Let product (any) soak 20 minutes. Don't squeeze mop the first time, spread it around. Second time get hot water again and mop it up