Chesshyre was a choral clerk of Trinity College, Cambridge during his time as an undergraduate at the college. From 1979 until 1993 Chesshyre was a member of The Bach Choir. Chesshyre now sings for the London Docklands Singers, which he joined in 2002. He has been, since 1980, a member of the Madrigal Society, the oldest musical society in Europe (see Madrigal). He became a Freeman of the Worshipful Company of Musicians in 1994 and a Liveryman of the Company in 1995.