It really depends on the civ you're playing. Most civs can play multiple openers well. Certain civs demand Tradition, like Ethiopia and Babylon. But others are more flexible.
Take Dido of Carthage for example. The free Harbor in every city almost begs you to take Liberty. If you go full Liberty/partial Piety/partial Exploration/Order, you can found an infinite number of cities even if you let them grow. (+1 happiness from Harbors, +1 happiness/2 cities, +3 production/coastal city, +1 production/city, fast-building monuments/temples/shrines, +2 happiness from monuments, +1 food shrine/temple, etc. etc. etc.... It's combolicious, and later in the game you're generating 300 faith/turn)
Add to that the ability to send production/food to your new cities, and it's just ridiculous.
And yet, the ability to capture cities early with Quinqueremes means you could easily win the game without ever founding a second city. So, Honor is a great choice. Those Quinqueremes can chew through barbarians and earn promotions, keeping your trade routes safe until they're +60% vs cities. :-D
Or go Tradition to get that one city moving really fast, because a tall production city can pump out Quinqueremes every other turn. You can take over the world with Tradition. Every city you capture has a harbor. On Archipelago or Continents that is... On pangaea if you want harbors you need to cap a lot of city-states. ;-)
Personally, I almost always go Tradition, but I occasionally open with Honor or Liberty. If I open with Liberty, I go for the free worker first (and I don't build one, because you get that worker at about the same time you'd have built it), then I go for the free settler, then I build 2 settlers, then I build the pyramids. Pretty standard, I think this is how most people do it. And it works. If you're having trouble making the shift from Tradition, remember this mantra: *You don't need to build that Wonder!*
I think when you always open with Tradition you get into this mindset where you think you *need* a tall city to do anything. But more cities = flexibility. You can still build that Wonder and build troops and buildings all at the same time. You can build cargo ships at the same time too, something *I* often end up forgetting to do when I go tall. because I'm chasing some stupid national wonder.
Liberty still works on the higher difficulty levels, but it's harder IMHO in BNW. You can't trade gold for luxuries out of the starting gate, so it's a slower start. Instead of 4 cities by turn 40, you might only have 2. But you can still hit 10 by turn 120. I think it now requires aggression though. You have to cap some of those cities. And eat the happiness hit by annexing them, because you can't afford for them grow. At least that's how I've been approaching it. I'm kind of a warmonger though. ;-)
EDIT: Also, for that Liberty approach I mentioned, pre-industrial units with faith is the key. When you go wide, you need to be able to defend any city without warning. Faith-buying units makes that a lot easier. Later on, the reformation belief to buy any great person with faith will give you something to do with all that faith, although finishing the Piety tree is often in conflict with Order by that point, so you may find yourself (as I do) with a lot of great prophets and nothing else. Eh, it's a work in progress.