Are there any devices that are like an iPod Touch, but run Android?
In other words, is there a piece of dedicated hardware designed (or shipped with) Android that is meant as a media player, so that I wouldn't have to pay for a phone, and just not use the phone features?
There's also the Archos 28 which is a little smaller and a little cheaper than the Archos 32. Again, they're both referred to as tablets but the Archos site (links previously) both say "Excellent multimedia player".
Also, MP4Nation may have some Android PMPs.
The Barnes & Noble Nook is based on Android, has no monthly fees, and includes the ability to playback music. It probably isn't what you have in mind as a media player, but I think it fits your description.
The Archos 32 sounds like it's the closest, so far. Although it is bizarrely referred to as a tablet.
Samsung has a Galaxy Player 50 PMP, which will be coming out soon.