At the main menu, press C-Up + C-Down to bring up the cheat screen. You'll see the message "You have found the secret room. Press some buttons and see what happens" will appear. Now enter the following codes at the Cheat screen:

Bonus teams:
C-Up, C-Up, C-Down, C-Down, C-Up, C-Up, C-Down, C-Down

Long necks:
C-Up, C-Up, C-Down, C-Up, C-Down

Big heads (home):
C-Up, C-Up, C-Down

Big heads (away):
C-Up, C-Up, C-Down, C-Left, C-Right

Big heads (both teams):
C-Up, C-Up, C-Down, C-Right, C-Down

Small players (home): 
C-Up, C-Down
    
Small players (away):
C-Up, C-Right

Small players (both teams):
C-Up, C-Right, C-Down 

Code         Player  
C-Down(3)    Kobe Bryant  
C-Down(4)    Ken Griffey Jr.  
C-Down(5)    Random NBA player  
C-Down(10)   Minoru Arakawa (Nintendo)  
C-Down(11)   Tim Bechtel (Nintendo)  
C-Down(12)   Steve Bolender (Nintendo)  
C-Down(13)   John Brandwood (Left Field)  
C-Down(14)   David Bridgham (Nintendo)  
C-Down(15)   Scott Bush (Left Field)  
C-Down(16)   Marc Doyal (Nintendo)  
C-Down(17)   Mike Fukuda (Nintendo)  
C-Down(18)   Jeff Godfrey (Left Field)  
C-Down(19)   Ken Griffey Jr. (Homerun Hero)  
C-Down(20)   Roger Harrison (Nintendo)  
C-Down(21)   Chick Hearn (Voice of the Lakers)  
C-Down(22)   Robert Hemphill (Left Field)  
C-Down(23)   Jim Holdeman (Nintendo)  
C-Down(24)   Mike Knauer (Left Field)  
C-Down(25)   Kevin Kraus (Nintendo)  
C-Down(26)   Chris Lamb (Left Field)  
C-Down(27)   Michael Lamp (Left Field)  
C-Down(28)   Stu Lantz (Lakers Announcer)   
C-Down(29)   Howard Lincoln (Nintendo)  
C-Down(30)   Ken Lobb (Nintendo)  
C-Down(31)   James Maxwell (Left Field)  
C-Down(32)   Umrao Mayer (Left Field)  
C-Down(33)   Arnie Myers (Nintendo)  
C-Down(34)   Dan Owsen (Nintendo)  
C-Down(35)   Colin Palmer (Nintendo)  
C-Down(36)   Tom Prata (Nintendo)  
C-Down(37)   Ed Ridgeway (Nintendo)  
C-Down(38)   Henry Sterchi (Nintendo)  
C-Down(39)   Noah Stein (Left Field)  
C-Down(40)   Faran Thomason (Nintendo)  
C-Down(41)   Gail Tilden (Nintendo)  
C-Down(42)   Russell Truelove (Left Field)  
C-Down(43)   Erich Waas (Nintendo)  
C-Down(44)   Phil Watts (Left Field)  
C-Down(45)   Armond Williams (Nintendo) 

Thanks to Revolution readers Daniel Dizio, Josh and Flyguy69!