I have used toilet duck for years, apply by cloth or sponge, leave overnight, then the following day, reapply toilet duck, leave for only about 3/4 hour, then using a scrubbing brush give the tiles and grout joints a brisk scrubbing, then rinse clean and dry, then I go over the lot again with moldex spray as it contains anti fungal and anti mildew properties tiles and grout should come up looking very good.
Or as suggested by Steve Cheevers, plenty proprietry brands avaiable
I do a lot of regrouting work, and as you say a grout rake is really no good, I use a fein multimaster, but there are plenty similar tools out there, such as the bosch pf 180, that will do the job, just buy decent blades, and the job is done fairly quick, however it can produce a bit mess lol.