I will attempt to say a person's name how they say it. If they say "call me [simple anglicized name]" then I will do that.
Most people seem to appreciate the effort. I'm sure my American vocals butcher some pronouciations, but I don't make a big deal out of it and my work gives me a lot of chances to practice. I will always make a good faith attempt at last names.
I live in Texas and will pronounce food names of Mexican origin with a Mexican accent. Burrito, taco, chalupa.
I grew up here, being white doesn't mean it's not my culture. I don't know jack about Mexican cuisine, but Tex Mex is my culture.