Sweeney guest-starred in the television series The Edge of Night and Spenser: For Hire. In films, he has played Jackie Willow, a Vietnam-era soldier, in Francis Ford Coppola's Gardens of Stone (1987), navigator 1st Lt. Phil Lowenthal in Memphis Belle (1990), and Travis Walton in Fire in the Sky (1993). He also played former ice hockey player Doug Dorsey in The Cutting Edge (1992), Shoeless Joe Jackson in Eight Men Out (1988), and Dish Boggett in Lonesome Dove (1989).
Sweeney was born and raised in Long Island, New York, by a Teacher father and a municipal government employee mother. He attended Shoreham-Wading River High School and both Tulane and New York University. Before completing his B.F.A. at NYU in 1990, he starred in the Broadway revival of The Caine Mutiny Court Martial.
He was a regular cast member on C-16 from 1997 to 1998.
After voicing Iguanodon Aladar in the 2000 Disney CGI animated film Dinosaur, Sweeney starred in Life as We Know It as Michael Whitman (2004–2005).
In April 2000, Sweeney married Ashley Vachon, they have a son, Cade (born August 26, 2001) and a daughter, Cody (born April 16, 2004).
Sweeney has guest-starred on various television series, including House (2006) as Crandall, an ex-bandmate of Dr. Gregory House; Jericho as John Goetz (2006–2008); Castle as a Los Angeles detective (2011), and more. He also had recurring roles as U.S. Marshal Sam Kassmeyer, assigned to protect Haley and Jack Hotchner on Criminal Minds (2009); as Peter Emory in season 1 of Crash (2008); and as Carter in The Event (2010).
A supporter of the U.S. military, Sweeney founded a website, lettersfromhollywood.army.mil to get messages of support to active duty and retired military. He has made multiple visits to the Middle East and Europe to visit with Service members. On May 4, 2008, Sweeney flew with the helicopter squadron HMM-364, "The World Famous Purple Foxes", in a CH-46E while in Iraq visiting with Service members and promoting his new film, Two Tickets to Paradise.
Sweeney plays Captain John Trent in the horror web series, Universal Dead. In late June 2010, it was announced that Universal Dead will be made into a feature film.
In 2012, Sweeney voiced the adult Avatar Aang in The Legend of Korra, the sequel series of Avatar: The Last Airbender. He currently narrates the reality television series Mountain Men.