This topic must be at the top of my list as personal pet peeves! Over the years as a Reading Specialist, I have heard the letter R mispronounced multiple times within my classroom, as well as on tv shows (reading specifically for kids), videos, and even online videos on teaching reading.
I think that one of the main reasons for this is that it is VERY difficult to pronounce correctly! Think right now of the word rabbit. Say it out loud- "rabbit." Now tell me does the sound the letter R says? Does it say /er/ (remember when you see // to say the SOUND of the letters inside)? No it does not.
Yet this is one of the most common errors people make when teaching this letter sound. The letter R makes a sound more similar to /ruh/ or /rih/; however, it's sound is often associated with the vowel which comes afterwards. For example, the R in run would say /ruh/, and the R in riff would say /rih/. However, if you were really going to be accurate about which sound R correctly makes you would have to remove all the vowel sounds....can you do that? Let's see. Say the sound /ruh/, now take away the /uh/. Difficult isn't it? Yet that is the correct sound for the letter R.
Author's Note: Now, I do not have a masters in Linguistics, so it is important for you to understand where my background for this information came from. It originally started within my linguistics courses at SDSU where we focused extensively on letter-sound correspondence (ie-phonics), then in my coursework for my teaching credential at SDSU, also within coursework for my Masters (with an emphasis in Reading), and most recently within courses taught by Houghton Mifflin.