I just figured this out and will post what worked for me--it's just a more detailed version of what macpos offered, because I had trouble finding the places he mentioned.
Open Keychain Access, which is in your Applications/Utilities folder. There should be two sections on the left sidebar, Keychains and Category. When I opened it, only the Category section showed, but I eventually found an arrow which expanded the Keychains section.
Choose Login in the Keychains section, and "Passwords" in the Category section. Use the search field to find the program you need the password for, and double-click on it. A window with that program's label will open. Under the Attributes tab, click on the little Show Password checkbox. You'll be asked for your admin password. Once you've entered it, the password will appear in the Show Password field.
As an aside, I thought this is what I was doing all along, so here's another thing I did before it started working: I found Keychain First Aid and ran both the Verify and Repair options. Both of these reported no problems, but it might have done something to encourage things to work right.*
*As you can tell, I anthropomorphize my electronic devices. It's the only way to cope with the way they misbehave.