You probably have rose slugs, which are little brownish green worms, sometimes they have a stripe down each side.
Neem oil is an organic pesticide and fungicide. I rarely use it summer months because it is also harmful to beneficial critters like bees, ladybugs and assassin bugs.
You can also spray the worms with a mild solution of castile soap and water. About a 1/4 teaspoon to 1 quart of water. The problem with that is munching on a leaf or stem that has been sprayed will not harm the destructive little monster, you must hit the worm in order to kill it.
Finally a very fine powder called Diatomaceous Earth will kill your invader when it comes in contact with it.
Although the powder is as fine as talc it acts like tiny razor blades with it comes in contact with any soft body insect. It's not harmful or abrasive to humans or animals. In fact some people use it as flea powder.
The only caution is not to breath the powder, it hard on the lungs, just like talc or or any other dust like substance.
I apply it to my rose bushes with a feather duster.
Diatomaceous Earth is available at most nurseries and some Lowes in the organic garden section.