Perspective taking is taking on the perspective of others. It's what we do anytime we buy a gift for someone else ("What would they like?"). So it means breaking the golden rule ("Treat others the way you want to be treated") and instead, acknowledges that others may not want what you want.
Even an organization that doesn't do much work internationally will benefit from a culturally intelligent strategy to innovation. Working across different generations, business units, regions, and functions are all factors that can also influence the innovation process.
When you jump on a city bus or roam the streets off the major thoroughfares you'll quickly learn that we have as many differences as similarities. And therein lie the greatest opportunities for innovation - our different ways of viewing the world and coming up with solutions!
Smaller companies are often more homogenous. Don't simply increase your diversity because of the social pressure to do so. Instead, realize that hiring a more diverse team will give you a whole new repertoires of innovative ideas. And then develop a strategy for effectively using the diversity of your team.
The distraction, particularly of technology, impedes the innovative process. And when you add to that the distraction of working with colleagues who are in different time zones and/or who have a different approach to urgency and distraction, the potential for losing focus is abundant.
There are highly innovative companies in the U.S., Germany, and India. And there are many stories of companies that failed to innovate in all those countries. This is good news because it means, regardless of your cultural background or where your company is based, it can become innovative.
Many Westerners see follow-through and reliability as the most critical factor in how they calculate the trustworthiness of another individual. In some other cultures, who you know and how you're related to other individuals is the most important variable. And for others, it may be as much about your reputation and what others have said about you.
I do think stories are one of the best tools for communicating across any number of cultures. But I also think there are wildly successful leaders who are introverted, disciplined, lead via spreadsheets and goals, and might not "appear" to be a great leader...but in retrospect, made a massive impact.
The two most important forms of diversity when it comes to innovation are visible diversity (typically skin color, age, gender, etc.) and underrepresentation (anytime someone is less than 15% of the majority group). Other forms of diversity are also relevant but these are the ones that psychologically play the most role in how someone engages with the innovative process.