I believe John Stockton holds the record with 19 straight playoff appearences, which also happen to be on the same team, Utah Jazz.
The coloumbus blue jackets never been to the playoffs but one
The Indianapolis Colts hold the record for the last decade, 9 appearances in the last 10 years. 2000, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009. They did not make the playoffs in 2001. If they make the playoffs in 2010, they will still have 9 appearances in the last 10 years, and will tie the record of the most consecutive playoff appearances in the NFL.
Through game 1 of the 2010 playoffs, Derek Jeterhas made the most postseason appearances. He has played in 139 games.
Dallas Cowboys, 9 total appearances in the playoffs
The current leader for most consecutive appearances is Arizona with 23.