Creating organizational cultures don't just happen by chance.
Indeed, the innovative folks over at 37 Signals argue that "You Don't Create A Culture" at all.
According to the article: "You don’t create a culture. Culture happens. It’s the by-product of consistent behavior. If you encourage people to share, and you give them the freedom to share, then sharing will be built into your culture. If you reward trust then trust will be built into your culture."
So, if you encourage and reward something (for example, an attitude or behavior) - on a consistent basis - it gets build into your organization's culture.
Sharing, trust, cooperation, engagement - whatever your organization has as its core values - must be encourage, recognized and rewarded, in order for it to become built into your culture.
Same goes for execution. If you value "getting the right things done" - consistently, predictably and in a balanced way, you need to encourage it, recognize it and reward it consistently over time.
BOTTOMLINE: Start today by creating your own "culture of execution". Determine what shared core values your organization has relative to getting things done, align your resources to do so, and begin to recognize and reward attitudes and behaviors (results, not just activities) that reinforce the kind of execution culture you want your organization to have.