You do not need ID to vote in Michigan. I have friends who have not brought their ID into a polling place in years, since this law went into effect, just to ensure the poll workers are not turning people away who do not have IDs.
If you tell them you do not have your ID, they are supposed to make you sign an affidavit attesting (under penalty of perjury) that you are who you claim you are, you are a US citizen, etc.