If the floor is big then a floor scrubber, although if the grout is well below surface level of tiles it will struggle.
As people have said toothbrush is good although will take some time.
If its just dirt then something like washing up liquid (mixed with water) will probably be fine, the action of the brush is doing most of the work not the chemical itself. The washing up liquid will just help break down a bit of grease and will act as a carrier for the dirt.
If the grout has been stained then experiment with a few things, Cif, white vinegar, washing up liquid, clothes washing liquid, diluted bleach or a bleach based kitchen spray etc
I would just start with some heavily diluted washing up liquid and see what that gives you and then decide if you need a deeper effect due to staining etc.