Mix one gallon of hot water, one cup of vinegar, and a few drops of dish soap. Mop the floor with the solution, using the scrub brush to remove stubborn dirt, or in high-traffic areas. Once the entire floor has been cleaned, mop once more using clean water to remove any remaining traces of baking soda or soap residue.
Bleach is safe to use for cleaning linoleum flooring, but you must dilute it with plenty of water. When cleaning floors with bleach and water, it's recommended that you mix ¾ cup of bleach with one gallon of water in a large bucket and use a mop to apply it to your flooring.