You made me curious as well. Part of the issue is that even among native English speakers, the pronunciation of R varies widely. A quick search of English R pronunciation on youtube turns up several different ways of pronouncing R, all by American speakers. The Irish, British, Australian, etc Rs are distinctly different as well.
The Old English R was trilled:
You'll hear the Old English R at the very end of this video, in the very last phrase:
I'm not sure exactly what happened in the development of the English Rs, but I guess there was some influence from the continent (mostly from French speakers) so the trilled/rolled R mutated, softened, and became the liquid consonant that is used today (in its varying forms).
Compare the first two R examples on this page and you'll hear how the second sounds like a lazy attempt at the first: http://en.wikipedia.org/wiki/R
(Even then, I don't pronounce my standard R in that way.)
I think the habit of "dropping" the R at the end of a syllable, as found in a number of British dialects (plus standard British English) as well as in Australian English, comes from the French and German Rs. English speakers probably found the throaty fricative R too rough and softened it until it became almost invisible. Standard British pronunciations of words like "beer" and "fear" will give you an idea of where that final R went.
Hope that helps a bit.