Bleach is a good cleaner to use no matter if you have used a natural cleaner or a chemical cleaner to get the initial shower grout cleaning done. Bleach will really penetrate the last of the remaining hard to clean mildew and mold out of the grout. It is not really wise to start cleaning with bleach because it does not have the scrubbing power that other cleaners have. Diluted bleach will simply break down those last to break down particles. Bleach will also kill much of the mildew and mold and will brighten up the grout so it sparkles white again.
Instead of making cleaning the grout a huge project that is only done once in a while the amount of cleaning that goes into the grout can be significantly reduced if a light periodic cleaning happens. There are shower tile sprays that can be sprayed onto the tiles after taking a shower that prevents the re growth of mold and mildew on the shower tile grout. Using these sprays every so often will really reduce the speed of the mildew growth and the amount of mildew and mold that actually grows back. These cleaners can be a huge preventative help especially for people that do not have a lot of time to clean on their hands.