It took a little longer than expected, but I've compiled what I believe to be a solid list of U.S. newspapers that offer RSS feeds. The list includes newspapers listed in the Yahoo! directory, including college/university publications when available. While it might be incomplete, it's the best list I've been able to put together based on resources (and patience) available.

Next steps: I plan on contacting the PR contacts at some of the larger newspaper companies and asking them if a) they have feeds available that haven't been found yet or b) when they plan, if ever, to implement them. Progress will be posted here or in a new entry on TMD. Thanks to everyone who sent me sites I didn't have yet or posted them in the comments - they were very helpful. [12/14: I think I'm going to make notations about many of the publications, to clarify them a bit, i.e. daily/weekly, etc.]

As always, any suggestions are greatly appreciated in the comments section or via email.

[December 23 - Indexed by State]
[January 2, 2005 - Added Detroit News]
[January 13, 2005 - Added Orlando Sentinel - [via Steve Rubel]
[February 23, 2005 - Added St. Louis Post-Dispatch and Star Tribune]
[March 5, 2005 - Added Des Moines Register and Milwaukee Journal Sentinel]
[March 13, 2005 - Added Terre Haute Tribune-Star]
[March 17, 2005 - Added Southwestern College's Collegian]
[March 26, 2005 - Added The Oklahoman]
[April 4, 2005 - Added Dayton Daily News]
[May 17, 2005 - Added AltWeeklies.com, Anchorage Daily News, Beaufort Gazette, Fresno Bee, Island Packet, Merced Sun-Star, Modesto Bee, News & Observer, Sacramento Bee, News Tribune (Tacoma, WA)]

[January 8, 2006 - Check out the official newspaper list here]

Alabama

Alaska
Anchorage Daily News
Petroleum News (on left side menu)

Arizona
Arizona Republic
Business Journal of Phoenix
Tucson Weekly (scroll to bottom)

Arkansas
Arkansas Democrat-Gazette (also includes: Northwest Arkansas Times, Benton County Daily Record, Siloam Springs Herald-Leader and WholeHogSports.com feeds)

California
Fresno Bee (direct link)
Merced Sun-Star
Modesto Bee
Sacramento Bee
San Diego Union-Tribune
San Jose Business Journal
San Jose Mercury News

Colorado
Denver Business Journal
Greeley Tribune (Colorado)
Pueblo Chieftain

Connecticut
The Day

Delaware
Delaware News-Journal

Florida
Jacksonville Business Journal
The Ledger
Orlando Business Journal
Orlando Sentinel
Tallahassee Democrat
Tampa Bay Business Journal

Georgia
Athens Banner-Herald
Statesboro Herald

Hawaii
The Honolulu Advertiser
Honolulu Star-Bulletin

Illinois
The Southern Illinoisan

Indiana
The Herald-Times (scroll to bottom)
Journal and Courier
NUVO (direct link)
Terre Haute Tribune-Star

Iowa
Des Moines Register (in left sidebar, News, Sports, Opinion)
Iowa City Press Citizen

Kansas
Wichita Business Journal
Wichita Eagle

Kentucky
Lexington Herald-Leader
Louisville Business Journal

Louisiana

Maryland
Baltimore Business Journal
Baltimore Sun

Massachusetts
Boston Business Journal
Boston Globe
Christian Science Monitor

Michigan
Detroit Free Press (scroll to bottom)
Detroit News
Lansing State Journal

Minnesota
Echo Press
Star Tribune

Mississippi
Jackson Free Press (direct feed link)

Missouri
Business Journal of Kansas City
St. Louis Post-Dispatch

Nevada

New Hampshire

New Jersey
Burlington County Times

New York
Business First of Buffalo
The Business Review
New York Times
The Village Voice

North Carolina
Business Journal of the Greater Triad Area
Charlotte Business Journal
The Daily News
The Herald-Sun (upper left corner)
The News & Observer
Triangle Business Journal

Ohio
Business First of Columbus
Cincinnati Business Courier
Dayton Daily News

Oklahoma
The Oklahoman

Oregon
Business Journal of Portland
The News-Review

Pennsylvania
Bucks County Courier Times
The Intelligencer
The Morning Call
Philadelphia Business Journal
Philadelphia Daily News, Inquirer
Pittsburgh Tribune-Review
Pocono Record

Rhode Island
Providence Journal

South Carolina
Beaufort Gazette
Island Packet
Spartanburg Herald-Journal (in left sidebar)

Tennessee
Nashville Business Journal

Texas
Austin Business Journal
Dallas Business Journal
Dallas Morning News
Denton Record-Chronicle
Galveston County Daily News (scroll down, left sidebar)
Houston Business Journal
San Antonio Business Journal

Utah
The Daily Herald

Virginia
The Free Lance-Press (Direct link)

Washington
The News Tribune
Puget Sound Business Journal
Seattle Post-Intelligencer
Seattle Weekly

Washington, DC
Washington Business Journal
Washington Post
Washington Times

Wisconsin
Milwaukee Business Journal
Milwaukee Journal Sentinel (direct link)

National Newspapers
USA Today
Wall Street Journal
AltWeeklies.com

College/University Papers
Auburn Plainsman (Auburn University)
The Badger Herald (University of Wisconsin)
The Battalion (Texas A&M)
The BG News (Bowling Green State University)
Brown Daily Herald (Brown University)
BYU NewsNet (Brigham Young University)
Campus Times (University of Rochester)
The Carolinian (USC Spartanburg)
Chicago Business (University of Chicago)
Chicago Flame (University of Illinois @ Chicago)
Chicago Maroon (University of Chicago - on right sidebar)
The Clarion (University of Denver)
Clarkson Integrator (Clarkson University)
Colgate Maroon-News (Colgate University)
College Heights Herald (Western Kentucky University)
The Collegian (Southwestern College)
Collegio (Pittsburg State University)
The Current (University of Missouri - St. Louis)
Daily Aztec (SDSU)
Daily Bruin (UCLA - scroll to bottom)
Daily Campus (University of Connecticut)
The Daily Cardinal (University of Wisconsin - Madison)
The Daily Free Press (Boston University)
The Daily Gamecock (University of South Carolina)
The Daily Iowan (University of Iowa)
Daily Orange (Syracuse University)
The Daily Princetonian (Princeton University)
The Daily Targum (Rutgers University)
The Daily Texan (University of Texas at Austin)
The Daily Utah Chronicle (University of Utah)
The Daily Vidette (Illinois State University)
The Dartmouth (Dartmouth College)
The Dickinsonian (Dickinson College - direct link)
East Tennessean (East Tennessee State University)
The Echo (University of Tennessee at Chattanooga)
The Greyhound (Loyola College)
The GW Hatchet (George Washington University)
The Heights (Boston College)
Iowa State Daily (Iowa State University - on right sidebar)
The Justice (Brandeis University)
The Kentucky Kernel (University of Kentucky)
The Lantern (Ohio State University)
The Maroon (Loyola University - New Orleans)
The Mass Media (UMASS Boston)
Michigan Journal (University of Michigan at Dearborn)
Missouri Review (University of Missouri - direct link)
The Monroe Street Journal (University of Michigan)
Mountain Echo (Mount St. Mary's College)
The Muhlenberg Weekly (Muhlenberg College)
The New Hampshire (University of New Hampshire)
The News Record (University of Cincinnati)
Northeastern News (Northeastern University)
Northern Star (Northern Illinois University - scroll to bottom)
The Observer (Notre Dame & Saint Mary's)
The Phoenix (Loyola University)
The Rebel Yell (UNLV)
The Seahawk (UNC Wilmington)
The Sentinel (Kennesaw State University)
The Snapper (Millersville University)
The Spectator (University of Wisconsin - Eau Claire)
The Sting (Southern Polytechnic State University - feed on right sidebar)
Student Life (Washington University in St. Louis)
Suffolk Journal (Suffolk University)
The Triangle (Drexel University)
Tufts Observer (Tufts University)
UCD Advocate (University of Colorado - Denver)
University Daily Kansan (University of Kansas)
The University News (University of Missouri, Kansas City)
The Villanovan (Villanova University)
Western Carolinian (Western Carolina University)

Thanks to Rabbit Trail for giving me a head start on this. Also, thanks to Derek Willis for some more papers.