This has already been done at Amazin Avenue, after originating at Brew Crew Ball. But it's interesting, so here it goes: the last trade the Minnesota Twins have made with every other team in baseball.
American League
Baltimore Orioles, September 5, 2008
Traded: Brian Bass
Received: PTBNL
Boston Red Sox, July 31, 2004
Traded: Doug Mientkiewicz
Received: Justin Jones (minors, from Cubs)
Chicago White Sox, September 3, 1986
Traded: Pete Filson
Received: Kurt Walker (minors)
Cleveland Indians, February 21, 1994
Traded: Enrique Wilson
Received: Shawn Bryant (minors)
Detroit Tigers, July 28, 2001
Traded: Mark Redman
Received: Todd Jones
Kansas City Royals, October 24, 1973
Traded: Ken Gill (minors)
Received: Tom Burgmeier
Los Angeles Angels of Anaheim, December 9, 2005
Traded: J.C. Romero
Received: Alexi Casilla (minors)
New York Yankees, August 31, 2003
Traded: Juan Padilla
Received: Jesse Orosco
Oakland Athletics, July 31, 2000
Traded: Mario Valdez
Received: Danny Ardoin
Seattle Mariners, March 29, 2008
Traded: R.A. Dickey
Received: Jair Fernandez (minors)
Tampa Bay Rays, November 28, 2007
Traded: Matt Garza, Jason Bartlett, Eduardo Morlan (minors)
Received: Delmon Young, Brendan Harris, Jason Pridie
Texas Rangers, August 25, 2008
Traded: Mark Hamburger (minors)
Received: Eddie Guardado
Toronto Blue Jays, July 16, 2003
Traded: Bobby Kielty
Received: Shannon Stewart, Dave Gassner
National League
Arizona Diamondbacks, March 28, 2001
Traded: Chad Moeller
Received: Hanley Frias
Atlanta Braves, July 22, 2001
Traded: Damon Hollins
Received: *Conditional Deal*
Chicago Cubs, November 13, 2007
Traded: PTBNL
Received: Craig Monroe
Cincinnati Reds, July 31, 2006
Traded: Kyle Lohse
Received: Zach Ward (minors)
Colorado Rockies, August 15, 2007
Traded: Ramon Ortiz
Received: Matt Macri
Florida Marlins, December 2, 2005
Traded: Travis Bowyer, Scott Tyler (minors)
Received: Luis Castillo
Houston Astros, August 3, 1999
Traded: George Williams
Received: Josh Dimmick (minors)
Los Angeles Dodgers, October 30, 1995
Traded: Kevin Tapani
Received: Ron Coomer, Greg Hansell, Jose Parra, Chris Latham
Milwaukee Brewers, November 15, 2002
Traded: Matt Kinney, Javier Valentin
Received: Gerry Oakes (minors), Matt Yeatman (minors)
New York Mets, February 2, 2008
Traded: Johan Santana
Received: Carlos Gomez, Philip Humber, Kevin Mulvey (minors), Deolis Guerra (minors)
Philadelphia Phillies, December 3, 2003
Traded: Eric Milton
Received: Nick Punto, Carlos Silva, Bobby Korecky (minors)
Pittsburgh Pirates, March 17, 1992
Traded: Denny Neagle, Midre Cummings
Received: John Smiley
St. Louis Cardinals, July 11, 2002
Traded: Warren Morris
Received: Seth Davidson (minors)
San Diego Padres, September 9, 2003
Traded: Todd Sears
Received: Alex Garcia (minors)
San Francisco Giants, December 15, 2003
Traded: Dustan Mohr
Received: J.T. Thomas (minors)
Washington Nationals, June 24, 2007
Traded: Levale Speigner
Received: Darnell McDonald
The A.J. Pierzynski-for-Joe Nathan, Boof Bonser and Francisco Liriano deal took place precisely one month prior to the Mohr/Thomas trade, which bumped that one off the list. Castillo to the Mets has almost been forgotten in the wake of Santana. Also, Brian Buchanon-Jason Bartlett trade happened a year before the Sears/Garcia pointless swap.
It might be interesting at some point to go through the trades team-by-team, to see who's gotten the best out of the team-to-team swaps over the years.