I happened to read something about touring Japan which warned not to "stray off the beaten path" not because its dangerous but because, appearantly, the Yakuza "import" these big Nigerians to basically hustle the obvious tourist into their clubs/hostess bars where they'll "pursuade" you into blowing all your cash.
Don't pay attention to stuff like that; it's crap.
Japan, and even Tokyo, is
ultra-safe, and probably one of the safest places in the world for straying off the beaten path; in fact you're probably safer if you don't look Japanese, because in some cases where someone would bother a Japanese, they'll leave you alone. Take advantage of that ultra-safeness and explore like crazy. [Another pro-random-exploration point is that Tokyo has so many train/subway stations, that you almost don't need to worry about getting lost -- no matter where you go, you'll eventually run into a station.]
In central Roppongi (but essentially nowhere else), you'll see big foreigners (indeed, mostly from Africa AFAICT) aggressively trying to lure you into seedy clubs, but you can easily just ignore them; they're annoying, but they won't do anything beyond constant whining. [Actually, there's been a crackdown in the last few years, so these days they've become almost timid.]
My advice: Just avoid central Roppongi; it's probably the tackiest and least interesting place in Tokyo. Problem solved.
But do explore. Tokyo is one of the coolest places on the planet.