Actually this is rather a doddle. I use Cif (Jif) Cream + a nail brush. Just wet the bar tape, apply the cream, and give it a scrub with the brush. 99% of the muck should lift off pretty sharpish, after which give the bars a hose down to get rid of any residue.
I am ultra-fastidious about clean bar tape but this little trick has meant that I've been able to keep running the same white tape for months on end.