Fill a spray bottle with a 50/50 mixture of water and white vinegar.
Spray the mixture onto the grout, saturating it thoroughly. Scrub with a wet nylon sponge with scrubber attachment. Continue until all grout is cleaned.
Sprinkle the sponge with a dusting of baking soda for heavier buildup as this gives an abrasive texture that helps break up the deposits. Scrub the rough side of the sponge over the grout and continue until all residue is removed.
Rinse off the shower grout and wipe clean with a dry rag. Also dry your shower after use, when possible, with a clean rag to reduce mineral buildup.