What kind of accents did "Americans" have around the time period of the American Revolution? Would they have sounded vaguely "British" to us? Had regional differences (Southern vs. Northern, for example) already appeared? Did they have their own slang or colloquial language that differed from the general British public?
The truth is that we don't know what people sounded like for certain because audio recordings from that period don't exist. The consensus, however, is that in 1776 British and American accents had not yet diverged. Both at that point -- we think -- were what are called rhotic accents (a person with a rhotic accent will pronounce the "r" sound in the word "hard") and it wasn't until after the American Revolution -- again, we think -- that the standard British accent (which British people would call a "southern accent," since it characterizes the speech of people in the southern part of England) became non-rhotic. Interestingly, the accents that characterized some of the Northern parts of the country like Massachusetts or New York became non-rhotic as well over time, while Southern accents became extra-rhotic and featured a lot of exaggerated "R" sounds.
I answered another thread about this. Allow me to paste the answer here.
In the late 1700s American English was just as different from today's as it was different from British English. "Though it would be clearly identifiable as English, it would be a variety of English unlike any we had heard before."
Words with a "kn" like knee would be pronounced like "t'nee" (only a generation before the founding fathers, Americans pronounced "kuh-nee")
The "ah" sound in words like father was not present. Instead, father would rhyme with modern lather.
Was was still being pronounced like "wass", but Kiss sounded like modern "Is" with the z sound at the end.
"Speech was in general much broader, with more emphatic stresses and a greater rounding of r's. ... Interior vowels and consonants were more frequently suppressed, so that nimbly because 'nimly' and somewhat was 'summat'."
You definitely wouldn't hear anyone say "thee" or "thou". That practice had been outdated for almost 100 years, replaced by the more polite "you."
I could go on and on about the pronunciation differences. The point is that they definitely did not sound British. My educated guess (I'm not a linguist or historian) is that it sounded more like the modern Boston/New England accent than anything else.
All this being said, you should know that, back then, the states were much more separated than they are today. The accent in Virginia would be very different from the accent in the Carolinas. Also, the speeches you hear today are taken from written text by these men.
Source: Made in America by Bill Bryson