I frequently ride across movable bridge grates in Chicago. Some of them have steel panels to make it easier for cyclists, but some do not.
Back in the day, I rode across these with 23mm tires. I don't ride anything that narrow anymore, but definitely haven't had any issues with 28-32mm road tires. I've never really worried about flatting, but the tires do tend to squirm back and forth a bit when you hit these, and it can be a bit unnerving.
Contrary to what others are saying here, I find that hitting these with some speed is the best way to go. I usually speed up and then stand and coast over the bridge, letting the bike move around underneath me as the tires kind of wander back and forth.
Here's a close-up shot of one that I took, the path of travel is right to left in this image, not top to bottom.
