There's tons of stuff of this nature, but most of it deals with the far future (Bebop/LoGH/most mecha/etc), an alternate modern world (Fullmetal Alchemist/Hunter x Hunter/etc), or archaic fantasy civilization (Berserk/Lodoss/Spice and Wolf/etc). If we don't include things like that, and look only at settings in 'our' world or close, with MCs that are not even ambigiously Japanese, then I can come up with:
Lupin III, Baccano, Jojo (only some parts), Rose of Versailles, Vinland Saga, Hellsing, Michiko to Hatchin, Thermae Romae, Ristorante Paradiso, 91 Days, California Crisis, Shuumatsu no Izetta
Basically anything from World Masterpiece Theater also fits your criteria, but these are based on non-Japanese works. There are other examples of non-Japanese stories as anime/manga, as well, like Gankutsuou, Buddha, or Takarajima.
Your mileage may vary with all of these, obviously.