I would 1st check w/the cemetery office. Some cemetery's have regulations on plant/tree size. At our family plot we had 2 arborvite on each side of the monument. They were there for years, the last time I saw them they were about 5'high. To my surprise on one of my visits, they were both cut down to the ground! I now have hostas planted in front. It makes it nice that for the most part something (the hostas) are always there.

Building on jrzgurl's advice, perhaps the cemetery would have some suggestions, or a nearby nursery. It's important to select plants that won't, in time, overwhelm the monument. I'd stay away from the bamboo, as most varieties tend to be pretty invasive ... you might like it, but your "neighbors" might feel otherwise.

Like jrzgurl, I found plants cut down completely. Finally decided to plant them in my yard instead - after all, my mom's not there in the cemetery ... she's in my heart and my everyday actions. That was the beginning of my gardening in earnest, resulting in beds and borders I'd never dreamed I could accomplish; appropriate, since Mom always wanted, but never had, flower gardens ... these are hers as much as mine.

There's plenty of plant material out there that is suitable - good luck to you in your search for the one that best suits your needs. It's so nice you're putting the thought and effort into caring for the site.