Mop your floor. I use a solution of 1 part vinegar to three parts water. You might not get the grout totally clean, but you need to start with a clean floor. Getting the surface grime off allows you to get the deeper stains off when you're really ready to scrub.

Make a paste with the lemon juice and baking soda. Lemon juice and baking soda are excellent natural cleaners for most everything in your home.[1] I'm not giving exact measurements because this depends greatly on the size of your tile floor, but you want your mixture to be thick and paste-like. Start out with a certain amount of baking soda (maybe a quarter cup and add tablespoonfuls of lemon juice until you have reached the desired consistency.

Spread the solution out on the grout lines. Let it sit for a few minutes, and then scrub it with the tile brush. A grout brush is a great tool for this because they have extra stiff bristles, but if you don't have a grout brush, an old toothbrush is a great substitute.

It's kind of funny that this is only one step because it's going to take you way longer than the rest of the process. You're going to have to really, truly scrub. If you're working on an entire room, expect your arms to be sore. Also, be sure to start at the back of the room and work your way to an exit; no scrubbing yourself into a corner, okay?

You're going to have to mop again. Sorry. But you just scrubbed sticky lemon juice all over the floor, and it's not like you can just leave it there. I use the vinegar/water solution as in Step 1, and make sure you get all of the residue up.