It must be a couple of years back that we had a discussion on this one, so it time enough to disinter it.

I'd agree with shona that "(n)eye-ther" is more prevalent in the UK, but (to continue the litotes usage ) "(n)ee-ther" is not uncommon.
And it seems to be based on personal preference, rather than on a regional or class base - the latter sometimes being a consideration over here, although much less so than in the past.