"What about Italian-American or Irish-American?"
I'm not sure. I never hear them, but it could be a regional thing. I hear people say things like, "my granny came from Ireland" or "I'm Italian" (meaning at least one of the ancestors came from Italy). Those terms might be more common in other parts of the country -- its a big country.
Those terms were never as problematic as "Mexican American." Mexican American was used to refer to people whose ancestors lived on territory that is now in the US and they NEVER immigrated. This is where the Day of the Dead connection is. And they weren't "Mexicans" so they aren't "Mexican Americans."
Mexican American was also used to refer to people whose ancestors immigrated from Mexico and people who directly immigrated from Mexico. It was also frequently shortened to "Mexican," which was problematic in a way that shortening "Italian American" to "Italian" isn't. (Mexicans drive across the border to shop, Italians don't).
It was also applied to people from throughout central and South American, which was just wrong. For example, referring to someone whose ancestors came from Peru as a "Mexican American". There are lots of immigrants / children of immigrants who are Latino/Latina, but not connected to Mexico in any way. Ironically, it has also been applied to people who are Mexicans and live in the US but aren't
Americans because they haven't been naturalized as citizens. I'm sure a lot of them celebrate Day of the Dead.
Additionally, here has been a really big mixing of people in the SW. Anglos and Latinos, Latinos from different countries, etc. One of the families I know that is really into the All Souls Procession the dad is from Peru, the mom is a pasty white chick, and the Day the Dead is big thing. But don't call the kids Mexican Americans, that's just ignorant. 