Hey Happy Camper,
I like your idea. I think that would really lighten the mood and create some motivation to speak up quickly. I like that your technique doesn't centre anybody out, so they all feel silly together and get rewarded for speaking rather than put on the spot.
I do something different with my shy class to get them to talk. I tell them it's
their turn to ask me questions (usually right after I take attendance, as a warm-up/review). I like to call this "game"
ASK ME QUESTIONS 
They know they can ask me anything, so the weaker students can get away with "How are you today?" and the less shy can try whatever new material we're learning, a more conversational "What did you do today?" perhaps. Sometimes I give points out for multiple questions and give a sticker for whoever asks the most.
(I don't do this in classes where there is one obviously unhindered student among the shy ones who will always "win".)
Putting them all in the spotlight at the same time, I'm freed up to do some prompting/hinting. I think it's empowering to give them the choice of topic to speak about, and I feel like most of my students don't have enough opportunity to ask questions, so it's good for practicing Qform. But the next time the minutes start to build up, I'll be sure to make them start walking.