To clean grout with baking soda, make a paste by mixing baking soda with water, and work the paste into the grout with a toothbrush or scrub brush. Mix a solution of 1 to 1 water and vinegar in a spray bottle, and spray the solution on the grouted area.
The vinegar causes the baking soda paste to bubble and begin to work on the soap scum or grime. After the paste bubbles, continue to brush until the grout is clean. Rinse with warm water to remove all of the baking soda and vinegar.
Another solution with baking soda that cleans scummy grout effectively employs bleach instead of water to create the paste. Use 3/4 cups of baking soda with 1/4 cup of bleach to make a paste as before. Apply this paste on the grout, and let it sit for up to 10 minutes. After 10 minutes, work the paste into the area by scrubbing thoroughly. Let it sit for an additional 10 minutes to let the paste do its work fully.
Rinse thoroughly, ideally with a handheld shower head if available. To avoid having to perform this thorough cleaning too often, keep a spray bottle with one part vinegar and one part water handy, and spray on the grout regularly.