If you try to force a strap that's too thick onto your Moto 360, you're gonna have a bad time
Yikes. What you see up there is my absolute favorite padded ostrich watch strap. It's the right size to fit the Moto 360 (that'd be 22mm, same as the LG G Watch), and I've been thinking of how good it would look once I got it on there. I figured I could squeeze it under the circular lip when I saw it was just too thick. The leather backing did not agree with me ...
The strap on the Moto 360 is easy enough to change. You do it like any other watch, whether it be Android Wear, or Pebble, or Timex. It uses the same spring bars many of us are familiar with, and the process is fairly simple. But you will need to make sure the strap you want to use is thin enough to fit under the little lip on the circular back cover.
Now Moto did tell us that they only recommend their leather straps. And they are very nice leather straps, indeed. I'll see if someone can't re-stitch the backing in place and put my favorite strap back on another watch. In the meantime, double check that your strap will fit deep into the lugs on the 360 before you get all hulksmash and stupid then try to force something like I did.
If you found a nice strap for your Moto 360, by all means take a picture and drop it in the forums with a link where you bought it! A lot of us will be looking for just the right one.