So I was going crazy thanks to the second-to-last bug on the Bugs and Problems wiki page: "Sounds cuts out after Press Start screen".  

Reinstalling the game didn't help, so I figured it was either some sort of VC++ issue, or something left behind in UserData. I deleted files one at a time until I discovered the culprit seemed to be the Reg0.dat file. I launched the game, it recreated the file, and sound worked again. Hooray!   So again:

• Close the game.
• Browse to the UserData folder (for the Steam version that's /Steam/steamapps/common/super meat boy/UserData/)
• Delete the Reg0.dat file.
• Relaunch the game, and sound should work again!

If someone feels like updating the Wiki page, that would be awesome.