Cancel the damned meetings. They have no point unless they have a point. I have been at too many of these over the last 15 years. I'm pretty sure that if the brain cells injured by these no-agenda "team meetings" were still functioning, my team and I would have built something that sold for billions of dollars and I could retire to relative obscurity and pursue my dream of fiddling with some code that only I care about for the next twenty years.
You probably have brief daily status meetings or stand-ups already. If you must continue the ritual of wasting a hour a week so that you feel like you're team building, then team build already. Set an agenda or request items from your team members that they think the team should address, and give them the power to go and act on those impulses after the meeting. That's it.
If there's nothing worthy of note on the agenda the day before the meeting, cancel it and go out to drinks (or coffee) for an early after work diversion, so that you don't interrupt whatever concentration might have actually been built up just before the scheduled interruption, and just talk about whatever comes to mind. Geek talk. It'll just happen. It's ok. If you're paying me well enough, I won't even mind buying my own gin and tonic.
I've had enough crappy lukewarm pizza at boring team meetings over the last 15 years of my career to add at least 10 of the excess pounds on my waistline, and I actually walk to and from work most days. I don't need it. Unless the people in my team suck, in which case I'm already plotting my exit, I'll build my own rapport, no structured exercises required. I'll get more done, I'll be happier, and my coworkers will like me more than if I feel obligated to present some attempt at proving my intellectual horsepower or listen to people drone on about process improvements or ritualized intellectual masturbation about source control provider factory seed data unit test models for widget acceleration and best practices for leveraging my strategic initiatives for performance test environments. (Sorry, I was snoozing and I just picked a random string of terms to regurgitate that I half-heard over the last twenty minutes of pointlessness.)