The vowel 'u' can be pronounced in many different ways, depending on the word and the position in the word.
I presume that you are not talking about words like 'up' or 'cut', or about words like 'uniform', which always begins with a 'yu' sound. I'm guessing that you are asking about specific words like 'tune', where the pronunciation varies. In standard British English, 'tune' is pronounced 'tyoon', whilst in standard US English it is pronounced 'toon'.
I can only answer for sure for the UK, but no, it's not a general rule. There are many regional and non-standard accents don't pronounce these words with a 'yu' sound. Many people with a typical London accent, for example, would pronounce 'tune' as 'toon'.
My guess is that most American accents would use the 'toon' version, with the possible exception of some pretentious-sounding Harvard-style pronunciations which might use the 'tyoon' version. But I could be wrong! Let's hope some US English speakers will be able to enlighten you on that one.