Willingham, acquired by Washington in a 2008 trade with the Florida Marlins, appeared in 114 games and hit .268/.389/.459 with 16 home runs and 56 RBIs for the Nationals in 2010, but ended the season on the disabled list after suffering a left knee injury that required surgery.
The 31-year-old outfielder now joins an Athletics outfield comprised of other 2010 acquisitions, David DeJesus and Hideki Matsui, as well as holdovers Coco Crisp, Ryan Sweeney and Conor Jackson.
DeJesus was traded by the Kansas City Royals to Oakland in exchange for Justin Marks and Vin Mazzaro earlier this off-season, while Matsui, a free-agent, signed with the Athletics this past Tuesday.
Heading to Washington in the trade for Willingham are Rodriguez, a right-handed pitcher who saw action in 29 major league games in 2010, and left-handed outfielder Brown, who spent 2010 in the minors, split between Oakland's AA and AAA affiliates.
In 27.2 innings pitched for the Athletics this past season, Rodriguez allowed 14 earned runs, struck out 33 batters and had an earned run average of 4.55.
Brown hit .320/.415/.502 with 10 home runs, 63 runs and 19 stolen bases for the Midland RockHounds, a Class AA Texas League team, but struggled in the Pacific Coast League while with Oakland's Class AAA club, the Sacramento River Cats, batting just .193/.253/.378 in 41 games.