It probably depends on where you are traveling and how remote you are going to be.
In general, I'd never leave cash on the bike...the bike is usually much too different from everything else out there...it's sometimes nothing but a beacon. If it got stolen or the bags on it stolen, I'd be SOL.
I have a small, waterproof shoulder bag/purse that doubled as a carrier for the camera, keys, small waterbottle, tourbook, cash, etc. It has a zip-close and a large flap over the top to conceal the zipper. The outside dimensions of the bag was a tad smaller than the inside dimensions of my handlebar bag. When I was on the bike, it fit nicely into the handlebar bag, and, I always had in my field of vision. When off the bike, I had it with me but didn't look like a tourist (except for the lycra shorts, helmet, and funny looking shoes
).
For my trip this summer, I had a back-up copy of my passport waiting for me in my email account in the form of a PDF file.
If I was to go somewhere known for pickpockets, I'd probably wear the extra cash + passport under my shirt or shorts, wear something bulky to conceal the lumps, carry the photocopies in the tourist book, and not let go of the tourist book.