I often hear Ashkenazim pronouncing (or at least trying) Hebrew words in Tefila/Keriat HaTorah like Sephardim. It probably isn't only Religious Zionists but it seems to be only them. Where did the this minhag come from? I heard live from either Rav Schechter or Rav Gedalya Shwartz that even the Rav (Solevecik) and Rav Kook would pronounce the Ashkenazi style. Maybe this came from Rav Nosson Adler (the Rav of the Chasam Sofer) who used to pronounce in the Sefardisha style? This is quite interesting also for some more info. http://www.dailyhalacha.com/displayRead.asp?readID=1778&txtSearch=pronunciation
My Hebrew teachers came from Israel, so they taught me how to read and write Hebrew as Israelis do.
I imagine those schools which do not hire Israelis to teach Hebrew, are also less likely to end up with students who feel a connection to religious Zionism.
However, I was also always taught that the Temani way of pronouncing Hebrew is actually the "most correct", and Israeli Hebrew is a mixture of all three, while also simplified to make teaching Ulpan easiest to the most people from the most amount of countries.
For the most part, the Sephardic pronounciation (with a few exceptions) is commonly accepted as the correct one.
That is to say, and I hope I'm not insulting anyone, much of what is known as Ashkenazic pronounciation (though not all of it) is actually a mispronounciation.
I say "commonly accepted", outside of the hardcore, ashkenoizis yeshivot (and except for non-bi-lingual Americans ;-)).
Then again, most Israelis speak broken Hebrew, anyway...
I will relate that I know someone, very learned and studied this specifically for a long time - in his tfilla, kriat hatorah, etc, he is very careful both to pronounce the ayin, chet, and other Sephardic elements, and also the saf, kamatz, and other Ashkenazic elements.
From personal experience:
My father pronounced Hebrew like a European, and my mother's father like his Gerer roots. But then we were sent to local day schools, many of which came into being during a widespread Zionist fervor (religious and not religious) in the middle of 20th century. There was a big push by the founders of these schools to institute a Sefardic pronunciation. While I ended up following my father, just about all of my classmates (and my brother as well) still use a Sefardic pronunciation to this day.