A family member told KOMO News that Nielsen's loved ones are grieving his death and would not have a public statement at this time.
Nielsen, 61, was rafting with his wife and some friends on the Green River east of Black Diamond on Saturday afternoon when his raft hit a patch of rough water and flipped over in the strong current.
Nielsen was thrown into the river's frigid waters, which are swollen with snowmelt this time of year.
The other rafters tried to reach him, but the current was too strong. Nielsen slipped out of his life jacket, and he was carried down the river, dragged along by the strong current.
His body was discovered at about 6 p.m. Saturday.
Nielsen's wife survived the rafting accident.
Nielsen initially was elected to the Everett City Council in 2004 and recently was re-elected to his third term on the council. He served on several committees.