There is a great deal of discussion about the correct pronunciation of the words during davening and krias hatorah. Many hold [and that is what it appears to this untrained non-linguist] that the Yemenites are the closest to the truth. They distinguish between every vowel and every letter [including significant differences between when there is a dot inside the letter or not].
I went to an elementary school [MDS] where our teachers taught us to speak in the traditional ashkenazi way - with it's pluses [e.g. distinguishing between tuf and suf] and minuses [not distinguishing between ע and א or כ and ח]. My parents are European born and also taught me the way they were taught by their parents and that is what I am most comfortable with.
In the modern schools they often teach the kids to say the words with the contemporary Israeli pronunciation [called הברה ספרדית even though it is not exactly the traditional Sefardi pronunciation] which is a bad mixture of some of the ashkenazi mistakes [ no difference between ע and א, כ and ח] and some sefardi mistakes [not differentiating between komatz and patach]. It is a language that was influenced by secularized Eastern European Jews but some people feel more Zionistic if they daven and learn that way. Others don't do it for ideological reasons but were just raised that way. It is not the end of the world but there is something to be said for preserving tradition.
When we speak on the street it would be weird to use "ashkenozees" but in shul it makes sense to keep tradition. The poskim generally say however that if one is raised using a certain pronunciation and changing it would make it difficult to concentrate he may continue as he is accustomed even if it is different than his ancestors. Otherwise, one should follow the custom of his parents. There is much literature on the topic. I direct you to a teshuva of Rav Ovadia [Yabia Omer 6/11] for a plethora of sources.
Rav Kook was considered quite the Zionist and he also encouraged people to daven the old fashioned way.
