Griemink barely played at all for most of last season in goal for Swindon after being replaced by Rhys Evans as first choice 'keeper.
Andy King released Griemink at the end of the season, and he is the first of the released players to ahve found a new club.
Southend have also signed another ex Swindon goal keeper as Richard McKinney who played for Swindon a couple of years ago has joined Southend from Colchester United.