Something is not right here. The timer shouldn't give a rats ass about ground. That's purely for safety. If it doesn't work without a ground connected, you've done something else wrong. What do you mean the grounds aren't connected to anything? Are they not connected to each other?

Is there not a white wire on the timer. What model is it?

Is your timer spring-wound or electrically powered? I suspect that you have a timer that requires neutral for it's own power and that the installation location has no neutral. The spring wound work just like a conventional switch for the electrical connections.

Here is a link that might be useful: Timer switch

The string wound intermatics don't tend to have leads (they have wire terminals either screws on the older ones or clamps on the later ones. The only ones that have leads all have a white lead which is necessary. He needs to tell us what model timer he is using.

That should say SPRING not STRING. The idea of starting a timer by pulling a string like you were starting the lawnmower is amusing.

The new timer is a GE Pushbutton Timer, made for lights or fans, and it says on the package "no neutral wire needed". It has black (hot), red (load), and green (ground) wires. In the wall I have wires for both the light and the fan switches. Two black go to the fan switch, white wires are just bundled together, and ground wires are bundled together (not connected to switches).

The instructions say the ground wire must be connected for the timer to work.

Is this the manual for the timer? (see link below)

Note this line:
Once power is restored it will take about 1 minute to fully charge. The timer will not function until it is fully charged. When charged, the 5 minute light will flash.

Did you wait a while before trying it out?

Here is a link that might be useful: 15318

Sound like a 'power stealing' device.

It may just be lightly capacitively coupled between the line in and ground.

Yes, that link does show the manual. I did wait at least a couple minutes each time I hooked it up to one of the ground wires. The first time, the fan turned on, like the instructions said, and turned off in a minute, but the timer lights never blinked so the ground connection must not be right.