Terry was born in Philadelphia, Pennsylvania, the youngest of three children. His father is Will Terry, an English teacher at the Germantown Friends School, which Michael attended. His mother is Holly Terry, a third grade teacher at Plymouth Meeting Friends School.
He studied cinematography and drama at Emerson College in Boston. During his time at Emerson, he made two student films: Blessed Is He and The Right to Bear Arms. In 2002, he apprenticed at the prestigious Williamstown Theatre Festival, where he worked on many main stage productions. Shortly thereafter, he moved to Los Angeles, where he worked with Noah Wyle's The Blank Theatre Company. He is also a member of L.A.'s the Brimmer Street Theatre Company. Terry starred in All Your Hard Work, written by Miles Brandman and directed by Michael Matthews in July 2012.