I have 18 different rose plants in my yard. They are my favorite of all plants and I really baby them. All of them have been planted within the last 3 years, so what I have to say can be taken with the ages of my plants in mind. After close observation this year, I can definitively say that only two of my roses have been TRUE continual bloomers - Double Knockout and Ducher.
My Ducher plant was planted last October, so it is quite a young plant. I think this is the reason it got a later start this year and didn't begin blooming till late April. However, since April I don't think there has been a single day when there hasn't been at least one bloom open on this wonderful shrub. It has only bloomed more as the year has progressed. No other rose can possibly perform any better in Louisiana conditions than this one. It's got a bushy, compact form, no disease at all, lemon-scented blooms, and it thrives with little care.
The Knockout family of roses is well-chronicled as an easy care bunch of roses with pleasing shrub form and continual bloom. My Double Knockout has definitely lived up to this reputation. It is NEVER without blooms, gets no disease, and certainly requires little care. I shear mine back every couple of months to keep it from becoming too big for its space. It has no fragrance at all.
A couple of my other roses get honorable mention for being very close to continual bloomers. Maybe when they get some more age and size they will become continual bloomers. They definitely have flushes of bloom every few weeks. These roses are Julia Child, Earthsong, Belinda's Dream, and Double Delight (grafted on Fortuniana rootstock.)