I thought it would be fun to go back and look at the longest All-Star droughts for a position from each team. I didn't consider DHs, since a DH isn't always selected to the All-Star Game, but did look at starting pitchers and relievers (although none of the 15 teams had those as their longest drought). You'll see that just because a franchise hasn't had an All-Star at a position for a long time doesn't necessarily mean they've had bad players there.
American League East
Baltimore Orioles: RF (Ken Singleton, 1981). Nick Markakis looked like he'd break the streak, but hasn't made an All-Star trip yet. In the '90s, the Orioles had a stretch of nine different regulars in nine years: Dwight Evans, Joe Orsulak, Mark McLemore, Jeffrey Hammonds, Kevin Bass, Bobby Bonilla, Tony Tarasco, Eric Davis and Albert Belle.
American League Central
Chicago White Sox: 2B (Ray Durham, 2000). Gordon Beckham looked like a future All-Star after his rookie season but has regressed. Tadahito Iguchi gave them a couple of good seasons, including the 2005 World Series year.
Detroit Tigers: LF (Steve Kemp, 1979). Bobby Higginson had some good years out there and Luis Gonzalez was the left fielder in 1997 -- but the Tigers traded him to Arizona (for Karim Garcia), where Gonzalez blossomed into an All-Star.
Kansas City Royals: CF (Willie Wilson, 1983). Just edges out second base (Frank White, 1986) and shortstop (Kurt Stillwell, 1988). Note: Carlos Beltran was traded in June of 2004 and made the All-Star team with the Houston. Close, but doesn't count.
Minnesota Twins: LF (Gary Ward, 1983). Position actually hasn't been that bad though: Dan Gladden, Shane Mack, Marty Cordova, Jacque Jones and Delmon Young all had multiyear runs out there and Josh Willingham won a Silver Slugger Award last year.
American League West
Los Angeles Angels: SS (Gary DiSarcina, 1995). First base has also had a long drought -- 1998, when Darin Erstad made it. As for shortstop, the Angels have only had six regular shortstops in the past 30 years: Dick Schofield, DiSarcina, Benji Gil (for one year), David Eckstein, Orlando Cabrera. and Erick Aybar. DiSarcina was probably the worst of the group but had a fluky .307 season in '95. Aybar has been a good player, averaging 3.1 WAR since 2008. Maybe he finally makes an All-Star team this year.
Oakland A's: 2B (Phil Garner, 1976). The A's haven't had a positional All-Star since catcher Ramon Hernandez in 2003, so they have several lengthy droughts. For center field and left field you have to back to 1991 with Dave and Rickey Henderson, and for third base to Carney Lansford in 1988 (Eric Chavez was never an All-Star). But Oakland second base gives us the longest All-Star drought of any AL position. The A's had some bad second basemen immediately after trading Garner to Pittsburgh (Marty Perez, Mike Edwards, Rob Picciolo), but for the most part have had decent second basemen like Mark Ellis, Tony Phillips and Mike Gallego and other veteran stopgaps (Joe Morgan, Willie Randolph). They've had 16 seasons of 2+ WAR from their regular second basemen since Garner.
Seattle Mariners: LF (Phil Bradley, 1985). Well, Mariners fans won't be too surprised about this. Ever since the M's traded Bradley left field has been a huge problem, other than those few solid years Raul Ibanez gave them. Even Mike Morse may only end up being a one-year stopgap. They've also had a long drought at third base -- Edgar Martinez in 1992. But Kyle Seager will end that this year.
Texas Rangers: 1B (Mark Teixeira, 2005). Josh Hamilton played more games in left field in 2010 and 2011, so he qualifies there. Before him, Juan Gonzalez was an All-Star in 1993 while playing the majority of his games in left.