I often find myself dipping into the same cultural cookie jar when drawing inspiration for writing. It doesn't matter what genre I'm writing in. I tend to fall back to Irish roots fairly easily, and focusing on Catholicism or similar religions. It's actually a bit strange. You see, I'm not Catholic. I don't doubt there are Irish roots in my family, however, but there are plenty of other roots too. There are bits of German, Jewish, Scottish, British and the other common European ancestry that I can draw from.
I can also draw from having a family that fought in the Civil War (both sides of the fence) and very well could reach to the American Revolution. My family has a broad swath of cultural possibilities.
That said, limiting yourself just to your own family isn't always the way to go. Limiting yourself to the cultures you know can be stifling.
I branched out from Europe and headed for Russia in response to stagnation and created something new. You see, that's one of the things I love about world building. You get a chance to write all of the histories, the religion, the culture. It's all up to you. That doesn't mean you can't take a hand up here and there. I drew inspiration from Russia, Ancient Rome, and even the beloved worlds built by others. I took the things I loved and hated, the things that married into unity, and created something that was a little bit of everything, but really it's own.
But that's really only one way to write using other cultures. Recently I was tasked with writing a horror story that did not stick to American convention. Something with new monsters. Something unfamiliar. Well, when I want the strange and unfamiliar, I dip into Japanese and Asian folklore. (They have the coolest monsters) The important thing is to do it right.
If you're going to write a culture with as much history and depth, you can't lose sight of what actually important to the story. You need to use the elements that are necessary to story telling and excise the rest. You can't be heavy handed with it. That's how you end up with Japan-o-phile sounding drivel and or some fanfiction.
Writing with another cultures roots should be done with respect, and a full battery of research, but don't let the culture outweigh the story you are trying to tell. Let it weave through the story.
I don't change the way I write horror when I write about things that take place in, or have characters from, Japan. My style remains the same. My characters are influenced by heritage, but unless for some plot reason, are not overwhelmed by it.
There are a lot of things you can do wrong when writing about another country's culture. You can do things wrong when writing about your own too. It's not mutually exclusive. Every culture should be treated with care. Research (Books people, not just the interweb) is your friend. You can, and should delve back as deep as you want. But never forget about researching the modern world as well. If you're writing a story that takes place in present day Dublin, for heavens sake people, RESEARCH IT! Get on googlemaps and learn the city.
Get on local websites, find out about local pubs and hotspots. Look at the news for the area. Really learn about the place where you've settled your story.
It can take hundreds of hours of research to write a book, it depends on your personality, how you research and how well you know your subject. That shouldn't dissuade you from trying, however. I enjoy research and also enjoy writing from prior knowledge. It's a balancing act.
So make fresh choices, set stories in new places and don't be afraid to try to write from another person's roots. Because limiting yourself can get you stuck. They say you should write what you know, in some senses that is true. But guess what, you can learn something new, and then write about that too. Just learn it first. It's a bit easier than researching half way through the book and having to rewrite. Trust me.