Many non-native speakers try to speak with an American accent by strongly pronouncing all the r’s. (Note that even the letter r itself is pronounced very differently by American and British native speakers.) Technically, the typical American accent is called a rothic accent, as compared to a non-rothic accent such as British English. A nice example is the word hard, which is pronounced as [hɑːd] in British English, but [hɑːrd] in American English, listen here. Leaving aside personal preferences (I much prefer a non-rothic accent, it is so much easier on the ears), it is important to realize that you cannot just put an r into any word to make it sound American. A good example I came across at scientific conferences is the word idea. The British pronunciation is [aɪˈdɪə], whereas the American pronunciation is [aɪˈdiːə]. However, what I heard was [aɪˈdiːər] (listen here), which is just wrong, except possibly on a pirate ship…
If you are interested, I came across this very useful post on how to pronounce r the American way:
Finally, there is a phenomenon called the ‘intrusive r’, referring to the insertion of r’s in order to avoid two consecutive vowels. See here for a nice discussion.