As you've probably discovered, grout is one of the most difficult surfaces to clean. It is both rough and porous at the same time. Rough surfaces are more difficult to clean than smooth ones, and dirt sinks into porous areas out of reach of your cleaning efforts. Moreover, grout lines themselves are recessed, so most cleaning tools glide right over them. Some cleaning tools actually make the grout dirtier by pushing dirt from nearby surfaces into the grout. This is especially true when you clean a floor.
A brush, along with the appropriate cleaners (Speed Cleaning™ Scum Bum in showers, Speed Cleaning™ Red Juice on counters) is the best way to get at the grout surface and dig out embedded dirt. Use a Speed Cleaning™ Grout Brush in a shower and, depending on the area to be cleaned, a toothbrush or a handheld stiff-bristled brush on a counter. When cleaning vertical surfaces such as shower walls, you can really put some muscle behind a brush, but grout on horizontal surfaces like countertops can be damaged by hard scrubbing. Grease, oil, coffee, or chocolate spilled on a counter isn't moved along by gravity to a drain. Instead, the grout becomes saturated. Many household products like grease weaken grout to the point that heavy scrubbing with a stiff-bristled brush will actually remove grout rather than clean it!
When cleaning countertop grout for the first time, go gently until you learn how strong or weak the grout actually is. If the grout has been soaked with various stains and has not been maintained well for a long period of time, you may not be able to get it clean again. Clean what you can without further damaging the grout. Let the cleaner sit on the grout for a bit longer and gently use a toothbrush or other brush. Ultimately re-grouting might be the best solution.
Protect new tile-and-grout countertops by being vigilant. Wipe up spills of all types—even plain tap water!—promptly. Note: a grout sealer can save lots of time by reducing the porosity of the grout. Apply it to new grout, and then reapply it every six months or so. On old grout or when reapplying sealer, first clean the grout as well as you can. Allow the grout to dry, then apply the sealer.