Is the rose a hybrid tea rose or floribunda? If so, the "suckers" are coming from the common rose used to hybridize the plant. They will not produce blooms, and for all practical purposes, the rose is putting all of its energy into this fast growing sucker... I would dig the rose up and start over. Many new hybrids are coming on the market that are "self-root", meaning they are hybridized but on its own stock, which will avoid the problem you may be having...
If, alternatively, your plant is an established rose bush that's not hybridized, don't cut the suckers, but rather, pull them off as close to the main stem as possible. It may be a climbing type rose attempting to do what comes naturally, in which case they should be ignored. (I would probably disagree with my friend Lonnie's contention re: leaves, but true suckers do have pale green leaves, green stems and few thorns.)...