I'm facing the same problem with one of my roses right now, Kelli. A huge Sally Holmes had to be moved a couple of years ago because of septic work. It apparently doesn't like the new location and started sending up suckers this year. Today I'm going to see how easily the suckers can be detached before the process starts to weaken Sally any further. Sally is down to only three or four canes of its own and now has four healthy suckers waving in the breeze. Today I'll dig down to pull the suckers off Sally and dig all around the rose to make sure the bud union is well above the ground. The person who moved Sally may have planted the rose too deep. In my garden I usually have no suckers as long as I keep the bud union at least 2 inches above the ground.