Easy! Unleavened bread was not used every week. It was the Festival of Unfermented bread also called Passover when it was made. And it was made in two batches during Passover week. One using old grain and another using new grain on the day of preparation for the last Sabbath of Passover. So unless you have something better than such get togethers that used bread, and do not specify unleavened bread you have no evidence. During Passover as in 1 Cor. this was understood as such requirements were well know by the reader. They knew Paul was discussing Passover and the special food requirements needed for it. The cup of blessing consumed during that meal shows this clearly.
Since the NT is scarce here, I think I will bring in some other info:
From the Didache, during the Apostolic era:
But every Lord's day gather yourselves together, and break bread, and give thanksgiving after having confessed your transgressions, that your sacrifice may be pure. http://www.earlychristianwritings.com/text/didache-roberts.html
Burn