In fact, there were several dialects of Hebrew throughout the centuries,
even contemporaneously. Indeed, what is thought of as the "Ashkenazic"
pronunciation is primarily a Lithuanian dialect which tended to become
increasingly commonplace (in no small part helped by the strong Lithuanian
yeshivot).
One opinion even suggests that each tribe had their own derech (witness the
Midrash in which each tribe had their own path thru the Yam Suf), and that
included a pronunciation. Thus, we had at least a dozen pronunciations,
even from the start.
The best answer to the matter I've heard comes from Rav Bulman of Ohr
Somyach in Yerushalayim, who, when asked about the matter, said that both
pronunciations are equally valid, and that one could use either (as was
appropriate to their background), but, he cautioned, "just don't make a
cholent out of it" (in other words, be consistent). The only limitation he
gave was that one must pronounce Hashem's name (i.e. the word Adon-oi or
Adon-ai as appropriate), according to his father's derech. In other words,
an Ashkenazi who learned Hebrew with Tavs and so on, can continue davening
that way, but should not start saying "Adon-ai" (the Sephardic pronunciation).
Frankly, I'm tired of the inferiority complex Ashkenazim are given about
their own history. Be fully aware that you are no less right than any one
else. Just because Zionists chose a different approach, most assuredly does
not invalidate yours. Be proud - you've got a lot of great people who spoke
the same way...
Eliezer Gamerman