'odd dashboard apps' would be any app that might announce something in the background. Dashboard apps can be running without you necessarily knowing, in the background.
Open up your dashboard (usually a click on the middle button or roller ball on the old Might Mouse) and see if you have any clock app that might be running and announcing the time or something.
Might also want to open system preferences for time and date and see if you don't have some option checked to announce the hour audiably. The sing-song sounds like one of the default voices too. Bruce maybe? Can't remember which is which.
While in your system preferences, go to Voices and check and see what voice you have, my guess is that it's the sing-song 'boring' voice.
Last thing to check, if there is not an option to announce time in time and date, it to Universal Access in System preferences. There is a whole host of choices for visual, audible and motor-skill handicaps and I would bet that announcing time would be one of them.