How d'ye do?

   This search turned up very few hits. It seems people all over Japan utters 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. Dôzo yoroshiku onegaishimasu.

 Aomori

   In Hachinohe

      はじめやんして

      Hajime yanshite

         (Possibly used throughout the former Morioka han, present-day Aomori and Iwate.)

Miyage

   ど～も～

   Dômô

   In Nishimorokata-gun

      はじめっじゃんそか

      Hajimejjannsoka

         (Standard Japanese: はじめてでしょうか)

   In Sendai

      おんや、まんずまんず

      Onya, manzumanzu

         (Can also be used when you receive something.)

Akita

   あら、初めでだんしな

   Ara, hajime dedannshina

   はずめでだんす

   Hazume dedansu

      (~だんす, dedansu is Akita-ben for desu)

   Southern Akita prefecture

      あったことねぇやな、はじめてだよな

      Atta koto nê yana. Hajimete dayona

         (A casual way of saying to a friend’s friend, “We haven’t met, have we? How d’ye do?)

Fukushima

   ちわ～

   Chiwah

      (Probably a contraction of konnichiwa) 

Chûbu - Niigata

   初だの～

   Not sure if this is read “Hatsu da noh” or “Haji da noh”, but it’s probably the latter.

   Sado-ben, spoken on Sado Island

      どこさんさぁ

      Dokosansâ

         (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?)

Aichi -  Nagoya-ben

      おみゃあさん、はじめてだなも

      Omyâsan, hajimete danamo

         (Apparently only older women use this phrase nowadays)

Kansai - Ôsaka

   まいど

   Maido!

      Seems like they say this a lot in Ôsaka.

Shikoku - Tokushima

   おうたことないんちゃう

   Ôta koto nain chau

      (Standard Japanese: 今までに会ったことはないでしょうか -- We haven’t met before, have we?)

Okinawa

   はじみてぃ、やいびーんやーさい

   Hajimichi, yaibîn yâsai

      (A casual, and rather long way of saying hajimemashite)