In -USA- we generally pronounce 'neither' with a long sound like [neether].

They're both acceptable pronunciations (although /ni:ðər/ is better than /ni:ðə/), and you'll likely hear both of them in all major English-speaking countries (to the best of my knowledge). I personally use both pronunciations whenever I feel like it, and they both sound perfectly natural. The same goes for "either". And yes, I'm aware of how weird that is.