I went through all the settings but couldn’t get it to sound.

Eventually I found the solution on this forum: when selecting what alarm sound to play, on the multiple choice menu,

A screenshot of the Alarm Sound Menu on my HTC Hero

use the volume control to adjust the Alarm Volume (this is different from Ringer Volume). Turn up the alarm volume to the desired level and test the alarm again.

Adjusting the Android Alarm Volume

It should work now.

This works for the HTC Hero, and I expect will be the same for other phones using the Sense UI. It may also work for other Android phones, based on my fiddling with the Android emulator. Does it work for you? Why not tell us what phone it worked for in the comments below.

EDIT: (13/03/2010 09:10) Made introduction a bit clearer about the nature of the problem (alarm goes off but without sound) and added final paragraph about what phones this may work for (potentially all Android phones if the emulator is anything to go by)