So what people are supposed to do is, if they want to pray, do so in private, but not have the assembly stop or begin with prayers?
Actually, assemblies
can be started with open prayers, but they must be general in nature and cannot be to a specific deity. Something in the nature of, "We pray for guidance from the divine in these proceedings. Amen" is acceptable. However, adding "These things we pray in the name of Jesus Christ, our lord and savior" or "Allah is great" or something similar is not. It's not that prayers are outright banned before or after the session, it's that Christians aren't allowed to invoke Jesus in the prayer that has them so up in arms. They claim it is a violation of their constitutional rights to forbid them from that when it has been ruled time and time again that it is not.
The city council of the city next to my hometown is going through this right now. One councilman is showing his ass and vows to continue fighting for the ability to pray to Jesus although he has been struck down at every step of the way. In fact, the US Supreme Court just recently refused to hear the case, meaning that he has lost and cannot pray in this manner. How he thinks he is going to fight beyond this is anybody's guess.