How d'ye do?
This search turned up very few hits. It seems people all over Japan utter the standard phrase “Hajimemashite” (初めまして) when meeting someone for the first time. That’s to be expected, I suppose. Why, even loquacious Americans can become rather stiff and uncomfortable around strangers. Dōzo yoroshiku onegaishimasu.
Tōhoku
Aomori
はじめやんして
Hajime yanshite
(In Hachinohe. Possibly used throughout the former Morioka-han (feudal domain), present-day Aomori and Iwate.)
Miyage
ど~も~
Dōmō
はじめっじゃんそか
Hajimejjansoka
(In Nishimorokata-gun, standard Japanese: はじめてでしょうか)
おんや、まんずまんず
Onya, manzumanzu
(In Sendai. Can also be used when you receive something.)
Akita
あら、初めでだんしな
Ara, hajime dedanshina
はずめでだんす
Hazume dedansu
(~だんす, dedansu is Akita-ben for desu)
あったことねぇやな、はじめてだよな
Atta koto nē yana. Hajimete dayona
(Southern Akita prefecture. A casual way of saying to a friend’s friend, “We haven’t met, have we? How d’ye do?)
Fukushima
ちわ~
Chiwā
(I suspect this is a contraction of konnichiwa)
Chūbu
Niigata
初だの~
(Not sure if this is read “Hatsu da noh” or “Haji da noh”, but it’s probably the latter.
どこさんさぁ
Dokosansā
(Sado-ben, spoken on Sado Island. Because it’s an island and everyone already knows everyone else rather than say, Nice to meet you, they ask where you’re from: どこの出身 -- Doko no shusshin?)
おみゃあさん、はじめてだなも
Omyāsan, hajimete danamo
(Nagoya-ben. Apparently only older women use this phrase nowadays)
Kansai
Ōsaka
まいど
Maido!
(Seems like they say this a lot in Ōsaka.)
Shikoku
Tokushima
おうたことないんちゃう
Ōta koto nain chau
(Standard Japanese: 今までに会ったことはないでしょうかーIma made atta koto ha nai deshōka?ーWe haven’t met before, have we?)
Okinawa
Okinawa
はじみてぃ、やいびーんやーさい
Hajimichi, yaibiin yāsai
(A casual, and rather long way of saying hajimemashite)