Cibocchi arrived at the County Ground last summer on a two-year deal and made a total of 18 league appearances, helping Paolo Di Canio's team win promotion as League Two champions.
He told the club's official website: "I would like to thank all the Swindon fans and everyone at the club.
"The fans used to sing 'dance with Cibocchi' but I cannot do that anymore! I have enjoyed my time with Swindon and wish them success this season.
"Also thank you to my team-mates for the past year I have had with the club because they have all made me feel very welcome and helped me."