Corruption in most of these countries is a negotiation game that is conducted implicitly. That's why playing dumb is so effective--if you don't know the rules of the game, you can't play, especially if you don't share a language. So, for every time where someone has forcefully pressed the point and made it clear to you that a bribe was expected, there have likely been a hundred times where a native in your place would have been expected to pay a small bribe of some sort.
And that really is the best defence.
It depends really much on how you travel. I don't know anything about you, but from the sound of it my guess is you backpack. If you're a backpacker, you probably don't have a problem as cops rightly don't see you as a good source of income. If you look really touristy you're probably fine as well most places.
If you're a resident expat, traveling in your own or a rented car you're right smack in their target market. And yes anyone can tell the difference between backpackers, tourists and expats. If you hang around in any expat watering hole you will hear all the worst stories (likely somewhat exaggerated by Rum).
It is perfectly possible to get through without bribing them in cash, and we did - we only ever gave out cigarettes. At one point though they did take one person from our group away from us, to a windowless cell with a chair in the middle, and interrogated him for a while. They were just trying to scare him into giving them money though, we managed to get out of that through a combination of persistence (both in not relenting, and pretending we didn't understand their broken English), name dropping ("I want to speak to the British embassy") and one of our group bursting into tears (no one likes dealing with someone crying, including Russian policemen).
I haven't been to Ukraine, but people I spoke to who took that route found it to be even worse than what we experienced in Russia.
Oh, and fun fact: The Western-most country where a policeman wanted a bribe was the Czech Republic. I was surprised!
I've also been told by a local to stick a $20 bill in my passport to avoid problems going through customs because we had expensive computer equipment on us and were in a rush for our next flight. I refused that advice. But I've seen boxes that have been opened and things missing as well so I was taking a chance by not doing it.
I've seen a lot of corruption, and had a chance to participate in it. Seems McAfee's advice is based on experience. I wouldn't discount it so easily.
Have you actually bribed a Canadian police officer?
I figured out after that the whole time I was being detained he was asking for a bribe (in a roundabout way, such that I didn't really understand with my poor sppanish skills).
You should an reddit AMA.
I started travelling a lot 5 or 6 years ago and have been to ~21 countries so far, and I don't travel every year. You could do 90 countries in say, six backpacking trips, each one to three months long. Doable in 3 years if you can keep it up. (that's barely getting to know each country, but still)
It is a small world: http://matadornetwork.com/bnt/seriously-slow-6-travelers-who...
I think McAfee's advice is way off.