I’m not just talking about accents like Irish or Scottish, though that would be useful too. I mostly mean if you’re not up to a road trip for example; how would someone from Ohio write a character with a New York way of speaking and not upset too many people?
Writing accents with weird spellings (“Ah cain’t buhlieve them thar cows jumped th’ faynce”) usually just makes the character speaking look silly, which can be fun for comedic effect sometimes but, like you said, can upset people (and also make dialogue a lot harder to read). The key to subtlety when it comes to portraying other accents in writing is in the vocabulary (local idioms and slang as well) and grammar, not the exact ‘sounds’ of the words. If you change any spelling, a little goes a looooooooooong way.
Take, for example, two ways of writing a sentence in an “Australian accent” (I chose Australian as I live in Australia):
In my writing, I do sometimes indicate a couple of spoken quirks via spelling, but it’s usually to show a character’s personality rather than their accent.