Assuming the 'dirt' is mold/mildew, open a window and try some full-strength bleach. Of course, wear gloves and watch out in getting it on your clothes. Wear your 'grubby' clothes. Put some bleach in a plastic container and use a toothbrush.
My best to ya and hope this helps.
Jay J -Moderator
PS: If you do this and it's still dirty, then the dark brown 'stuff' isn't mold/mildew, but dirt instead. Try and rent (or buy) a small, hand-held steamer. The steamer will need a little pressure. In any event, when you're finished cleaning the grout, kinda wash it down with fresh water and let dry for a few days. Then seal with Grout Sealer.