Nope. Use abstract facing, just like AD&D. You just need to keep track of whether you're moving at full speed or half speed this turn, and that will tell you whether or not you're going to take opportunity attacks.
If someone says, "Why can't I just back up 10 feet and then turn around and move at full speed," the answer is, "We're not using facing rules. You can try that, but if you do it's equivalent to moving forward, and you take an opportunity attack when you back up. (You're distracted planning your movement.)"
However, you can back up 10 feet, turn around, and attack another guy five feet away, if the first opponent didn't follow you when you moved. To me, that's a plus and perfectly realistic. The system is working as intended.
Edit: I should mention that this is not really so much about which direction your physical torso is facing, more about what you're paying attention to. I don't see any problem with the fact that under these rules, being surrounded but "unmoving" doesn't grant opportunity attacks.
Edit2: I also like the fact that under these rules, creatures with all-round attention such as black puddings and beholders will not take any opportunity attacks no matter how fast they move, because they have eyes all over. Everything just fits together and makes in-world sense, without needing to add finicky abilities like Nimble Escape to those creatures as special cases.