There have been 14 players in NBA history that have represented 3 different franchises in the NBA Finals. John Salley was the first player to win a championship for all 3 (89-90 Pistons, 96 Bulls, 00 Lakers). Three of the other players have since joined him -
Robert Horry (94-95 Rockets, 00-02 Lakers, 05 and 07 Spurs)
LeBron James (12-13 Heat, 16 Cavs, 20 Lakers)
Danny Green (14 Spurs, 19 Raptors, 20 Lakers).
Some others on the list weren't so lucky.
Larry Foust went to the Finals in back-to-back seasons with the 1955/56 Fort Wayne Pistons, coming up short against the Syracuse Nationals and Philadelphia Warriors. He later got three consecutive cracks against the Bill Russell Celtics dynasty from 1959-61, first with the Lakers in their final season in Minneapolis, then with Bob Pettit's St. Louis Hawks, but was again unsuccessful.
Max Zaslofsky is the only player whose last name starts with the letter "Z" to make an NBA Alll-Star team. Unfortunately, he is also 0-4 in the NBA Finals, despite getting chances with three different teams. He was on the losing end in the very first NBA Finals, as his Chicago Stags fell to the Philadelphia Warriors. He would later join the Knicks, losing two 7-game heart breakers in 1951 and 1952 to the Rochester Royals and Minneapolis Lakers. In his last Finals appearance, he would suffer the same fate as the aforementioned Foust with the Pistons against the Nationals.
Gary Payton, Sam Perkins, and Eric Snow all went to exactly three Finals, with different uniforms on each time, the only three players in NBA history to accomplish this rarity. Payton would manage to get that elusive ring with the Heat, but Perkins and Snow weren't so lucky. Also, in an amazing coincidence, all three players were teammates on the 1996 Seattle SuperSonics.
Payton (96 Sonics, 04 Lakers, 06 Heat)
Perkins (91 Lakers, 96 Sonics, 00 Pacers)
Snow (96 Sonics, 01 Sixers, 07 Cavs)
The remaining five players on the list:
Danny Ainge, 6 appearances (84-87 Celtics, 92 Blazers, 93 Suns)
Wilt Chamberlain, 6 appearances (64 Warriors, 67 Sixers, 69-70 and 72-73 Lakers)
Shaquille O'Neal, 6 appearances (95 Magic, 00-02 and 04 Lakers, 06 Heat
Clyde Lovellette, 5 appearances (54 Lakers, 60-61 Hawks, 63-64 Celtics)
Kendrick Perkins, 4 appearances (08 and 10 Celtics, 12 Thunder, 15 Cavs)
Steve Mix is not on the list, but this is some bad luck worth noting. Mix played 9 seasons with the Philadelphia 76ers from 1974-82, appearing in three NBA Finals but coming up short all three times. In the 1983 season, he signed with the Lakers just before the playoffs, who had ended the Sixers' two previous Finals bids. And, as luck would have it, 1983 would be the year the Sixers finally figured it out, and Mix would go home ringless once again.