Any evergreen groundcover will work as a living mulch. Thyme needs full sun and excellent drainage to thrive and is also quite low growing, so won't effectively "hide" what your doggy leaves behind :-)

Here is a discussion of a few that would work well in a CT garden. But I wouldn't limit myself.........check out your local nurseries when they open for business and see what's on offer. Whether or not you will actually need to remove the mulch before planting will depend on the type and thickness of the mulch and the type of GC you choose ;-)

Groundcover plants

check this one out:

http://www.finegardening.com/plant/red-epimedium-red-barrenwort-bishops-hat-epimedium-x-rubrum

Look at ajuga. I usually dislike it as a groundcover since it is very aggressive when happy, and low enough that it can't be controlled by a lawnmower. However, if you don't have a lawn to keep it out of, that isn't an issue. There are foo-foo cultivars out there, but the usually purply type is indestructable.