Write the assigned verses or passages on a post it note. Mix up the order you will be covering the passages in, and stick an assignment under each seat in the classroom. When it comes time to read, have everyone look under their seat for their passage and read.
I usually employ this method when we have several scriptures to read from different areas of the scriptures.
When using this method -- especially with adults -- be sure that everyone in your class can read before class begins. If you know some members of the class are not thrilled with reading aloud, remind the class that it's okay to mispronounce words or read slowly. I do it, too, and Church is a judgment-free zone.