Here are the five players, two forwards, one center and two guards, who fell just short in my opinion. Note that from here on out, you can assume that the years listed are gold medal-winning years unless otherwise noted.
SF: Larry Bird (1992)
SF/PF: Carmelo Anthony (2008, 2012, 2004-Bronze)
C: Bob Kurkland (1948, 1952)
SG: Adrian Dantley (1976)
PG: Jason Kidd (2000, 2008)
Bird, though the mascot of the 1992 Dream Team, doesn't quite have enough talent to make the final 12. By 1992, he was battling injuries and only started three of the eight games in Barcelona.
The thrice-capped 'Melo falls to the honorable mention slide due to the immense depth Team USA enjoys at the forward position.
You've probably never heard of Kurland (I hadn't either; he never suited up in the NBA). But he was one of eight Americans to receive consideration for the FIBA Hall of Fame.
A 7-footer, Kurland won two gold medals as an AAU representative on the 1948 and 1952 gold medal-winning teams, partially due to the fact that he was one of the first players to use dunks as a consistent part of his repertoire. He also was a six-time AAU All-American and made the Naismith Hall of Fame in 1961.
Adrian Dantley, who later became a Rookie of the Year with the Buffalo Braves and a scoring champion with the Utah Jazz, led the gold medal-winning Montreal team in scoring with 116 points in six games.
Jason Kidd has the distinction of starting the most games at point guard for Team USA (eight in 2008, plus four in 2000). Kidd also ranks in the top 10 of Team USA players in career rebounds and assists.