This was not unique to my grandparents, of course, or even to other Italian immigrants of that era. Immigrants from everywhere were bringing their home cultures and traditions to the United States. They still do.
I mention this here because my personal preference for talking (or writing) about Italian food is in the context of the "old country". Another phrase I like to use to refer to Italy 100 years ago is "classic Italy". In classic Italy (i.e., the old country, or Italy 100+ years ago), life was different than it is today. Actually, life was different everywhere 100 years ago, but we're focusing here on Italy!
In classic Italy, and especially in rural areas, meals were prepared using only the ingredients that were available locally. There were no distribution systems in place; and even when foods from other regions were available they were too costly for all but very special occasions.
Consequently, olive oil was a common ingredient in recipes originating in southern Italy (where most of the olive groves were found), but much less common in recipes from central and northern regions. Similarly, pork fat, lard, milk, and butter were common ingredients in recipes originating in central Italy, in the Emilio Romano region, where pigs and cows were raised, but less so in southern recipes. Tomatoes, especially the San Marzano tomato grown in the lava-rich soils surrounding Mt. Vesuvius, were a primary ingredient in many (if not most) southern Italian recipes.
All that has changed, of course. Italian foods and ingredients are now generally available everywhere – even the United States! As of May 28, 2013, the FDA lifted its four-decade ban on importing (most) Italian cured meats. (Before then, only mortadella, prosciutto di Parma, and prosciutto di San Daniele were permitted.)
Most important, however, the classic recipes have survived – their origins unchanged. This provides us the opportunity to recreate and enjoy authentic, classic, dishes that Italian grandmother’s would have served their families in the old country, in their own, individual style.