Palmieri and Cizikas each was an unrestricted free agent. Beauvillier and Sorokin were restricted free agents.
Palmieri scored 21 points (10 goals, 11 assists) in 51 regular-season games with the New Jersey Devils and Islanders last season, including four points (two goals, two assists) in 17 games after he was traded to New York, along with forward Travis Zajac, on April 7.
The 30-year-old forward scored nine points (seven goals, two assists) in 19 Stanley Cup Playoff games last season to help the Islanders reach the Stanley Cup Semifinals.
Selected by the Anaheim Ducks with the No. 26 pick in the the 2009 NHL Draft, Palmieri has scored 359 points (185 goals, 174 assists) in 612 regular-season games with the Ducks, Devils and Islanders, and 24 points (15 goals, nine assists) in 57 playoff games.
Beauvillier scored 28 points (15 goals, 13 assists) in 47 regular-season games and 13 points (five goals, eight assists) in 19 playoff games last season.
The 24-year-old forward was selected by the Islanders with the No. 28 pick in the 2015 NHL Draft and has scored 155 points (81 goals, 74 assists) in 333 regular-season games and 29 points (15 goals, 14 assists) in 49 playoff games.
Cizikas scored 14 points (seven goals, seven assists) in 56 regular-season games and five points (two goals, three assists) in 19 playoff games last season.
The 30-year-old forward was selected by the Islanders in the fourth round (No. 92) of the 2009 draft and has scored 185 points (81 goals, 104 assists) in 590 regular-season games and 15 points (five goals, 10 assists) in 69 playoff games.
Sorokin, who made his NHL debut Jan. 16, was 13-6-3 with a 2.17 goals-against average, .918 save percentage and three shutouts in 22 games (21 starts) in the regular season and 4-1 with a 2.79 GAA and .922 save percentage in seven playoff games (five starts).
The 26-year-old goalie was selected by the Islanders in the third round (No. 78) of the 2014 NHL Draft.