Alexander Duncan, 59, was found in a flat in Primrose Street on Sunday.
Police Scotland is appealing for witnesses who may have seen anything "unusual or suspicious".
Det Insp Martin MacLean, of Police Scotland, said: " I am keen to speak to anyone who was in the Primrose Street area from Thursday 16th April onwards or who knew Alexander Duncan."
The death of a man whose body was found in a Leith house is being treated as murder.