Non U-Lock that can be mounted for carrying? I recently moved away from NYC to the suburbs. In NYC, I always locked up my bike with the New York Lock, which is a super heavy duty U-Lock. With a plethora of signs and parking meters, NYC works amazingly well for a U-Lock. Now however, the security is overkill and also there's very few signs and no parking meters in the suburbs. I find that I spend at least 10 minutes when I get places trying to identify a U-Lock compatible location to lock up.

I'm looking for a bike lock that isn't a U-Lock but is still fairly secure. I might not need such high security still, but you know the old saying... (You can take a man out of the city but you can't take the city out of a man.)

My one requirement is that whatever chain/cable lock I end up with needs to have a mount for the bike. I don't want to be carrying this or draping it on the bicycle.

I've found tons of cheap, really insecure locks with mounts. I can't find anything that's both secure and mountable.

Cost is no obstacle here. A bike theft would cost much, much more.
Any specific lock recommendation is going to be out of date in a year, so I'll focus on identifying the best kind of lock: 

Any cable locks are going to be insecure by nature, but it's unfortunately the only other kind of lock available just now. (I'm not counting big-ass chains with giant locks; those are more for show than for security.) 

One of the heavier Kryptonite cable locks is probably your best bet. To beef up the security, I'd get two of them and lock up different parts of the bike. I'm told that keyed locks are harder to crack than combination locks. 

However, have you considered getting a smaller U-lock and a long cable? You'd get the best of both worlds that way. Since places to lock up are scarce, you'd usually end up using it as just a cable lock, but it'd give you flexibility in the event you find a good sign or bike rack. 

Bike mounts: These tend to fall apart rather quickly. I recommend focusing on a small pag to carry your locks, rather than use a frame mount. There are triangle frame bags and small trunk bags that do the job nicely. However, cable locks have a great advantage over U-locks--you can coil them up and hang them from your handlebars. Unless you have smaller drop bars, it's amazingly convenient.