You can count reliever Doug Jones among them.
An All-Star with the Indians during his first stint with them from 1986 to 1991, Jones returned to Cleveland last week in a trade that sent middle reliever Eric Plunk to Milwaukee.
"I vaguely remember those years and not very often," said Jones. "But the Indians gave me my first real opportunity to play Major League Baseball for any length of time. I played with a lot of very good players, and I always felt like I was the rookie in the clubhouse."
Jones can be forgiven if he can't believe this is the same Cleveland club that let him go after the 1991 season, in which the Indians set a team record for 105 losses in a season. Jones still holds the team's career record for saves with 128.
Jones, a reliever with a tricky change-up and a less than intimidating fastball, played before sparse crowds and cool lake winds at Cleveland Stadium on teams that consistently dwelled near _ or in _ the bottom of the AL East.
Since his last season with Cleveland, the Indians have been to two World Series, won three AL Central division titles and made sellouts at Jacobs Field routine. Many fans now consider it a question of when, not if, the Indians will win a world championship.
But Indians' fans have long memories and haven't forgotten about Jones who was often a lone bright spot for Cleveland.
Jones received a warm welcome when he made his way to the mound in the eighth inning of the Indians' 8-1 loss against Detroit Sunday.
"It was really weird having people cheer in Cleveland," Jones said. "It was quite a moment. It really got me charged up."
Jones was an All-Star with Cleveland but Indians General Manager John Hart believed Jones' best years were behind when he cut him loose after the '91 season.
Jones' post-Indians career took him through Houston, Philadelphia, Baltimore, Chicago (Cubs) and Milwaukee. He saved 162 games along the way and returned to the All-Star Game last season for Milwaukee after posting 36 saves and a 2.02 ERA.
Jones has slipped some this year, blowing 8-of-20 save opportunities while posting a 3-4 record with a 5.17 ERA, and giving the Indians a chance to get the closer. Tribe coaches think they have detected a flaw in Jones' delivery that can be corrected.
Jones' return to Cleveland was part of a major shake-up in the bullpen. Cleveland acquired pitcher Steve Reed and outfielder Jacob Cruz from San Francisco for troubled closer Jose Mesa, infielder Shawon Dunston and pitcher Al Morman.