Actually, once upon a time, *I* had that first-generation accent thing.
I used to get mistaken for a very-well assimilated nonnative English speaker by random Europeans all the time. (Americans would assume I was a Brit who had been here a long time.) My grandfather, who was my primary childcare between the ages of on, 18 months and 6 years, was a Swedish immigrant--he immigrated at the age of 12, and had no detectable accent.
What he did have was a very crisp and technically perfect pronunciation. (There was no "gonna" in that house. On pain of Disappointing Papa.) And the use of things like "moment" instead of "second"--he spent WWII in England, which also affected his speech. And my grandmother's father was Irish, and she got some of that too.
Ah, hyperliterate English! The baseline speech of a certain sort of online community. (Although I notice that my effort to colloquialize my speech and writing has made the literary/literate version harder to produce.)
My Yankee only ever came out in in "kwadah" (one fourth of something) and "wadah" (which you drink.)
Seven years in Las Vegas and the ceaseless mockery of my co workers, alas, ruined that. But it couldn't get rid of my kwadahs.
I notice one of my best friends, who is American born and has a Scottish-Indian mother (raised in Scotland) and a south Indian father, both of whom have kept their accents--there are certain words where you can pick out the duelling parental accents, and otherwise she's bog-standard swamp Yankee. (She uses "baahsil," not "bayzul," to season her food, for example.)
I think Trinker sounded like a Californian to me, but it was a looong time ago, and I am not as adept at picking out linguistic influences as a trained phonologist--or the Car Talk guys, who can tell you everywhere you've lived and where your parents are from, half the time.
My English at the time we last had any face-to-face, Bear, was probably as Californian as it has ever been, living fulltime with two Angelenos.
My poor eldest is mostly me-inflected, with bizarreties like "owe-vn" (oven) because of Gamma. Thankfully, he doesn't have my kid brother's mushmouth.
One of the most incomprehensible accents I have heard was on a Cockney who moved to rural Virginia and lived there for twenty years.
Ooh, that's a good one. I'm still fond of the mix I heard in Chinatown of either Bronx or Brooklyn overlaid on Cantonese. That was awesome.