Ivy lives to climb and that is true for vines. What you need is a ground cover that is not a vine.

Dwarf mondo grass: Ophiopogon japonicus 'Nana'. It looks like miniature liriope but it is not the same plant at all. Doesn't climb and is not aggressive like Monkey grass (liriope). These clumps will eventually grow together.

{{gwi:810007}}

If you want something that has more of the trailing ivy look to it. there is vinca - both minor and major. However, unless you have a concrete barrier - like a sidewalk - around it, it spreads everywhere. I have vinca minor in front of my house. It's bordered by the house, the sidewalk, and the driveway, and it can't escape. If you are looking for something to cover a huge area, I like the look of vinca major - again, you need a serious barrier to keep it out of areas where you don't want it. I love the vinca major under the magnolia trees on the UGA campus.

Hi, We're looking for something more along the lines of the vinca mentioned looks wise, only not so spreading that it needs a concrete barrier...

Creeping Rubus

http://www.ces.ncsu.edu/depts/hort/consumer/factsheets/groundcover/rubus_calcyinoiden.html