Can I put a 6 foot privacy fence on my property line in Macomb Township?

Generally yes, on rear and side lot lines. Macomb Township's fence ordinance allows privacy fences on or parallel to rear or side lot lines at up to 6 feet (Code of Ordinances Sec. 14-26.d.1.a), and the location rules actually direct that fences be placed on the lot line where practical, with the outside edge on the property line (Sec. 14-27.a.1). The finished side must face your neighbor or the street (Sec. 14-26.e).

Limits to watch: the fence may not extend into the front yard past the front building line (Sec. 14-27.a.3.a); no fence may stand within an open county drain easement, and fences abutting one go on the easement line unless Macomb County Public Works approves otherwise (Sec. 14-27.a.5); and double fencing is prohibited, so you cannot build within 3.5 feet of an existing fence except where the building official permits parallel fencing (Sec. 14-27.a.6). A permit from the Building Department is required before you build (Sec. 14-25).