I'm not sure why, but there are many cichlid enthusiasts, especially novices, who question the compatibility of cichlids and live plants. Anecdotal evidence suggests the controversy about this revolves around the decision to emulate a specific biotope. Trying to exactly match the natural habitat from where your cichlid species originate is really unnecessary and practically impossible. By matching the habitat, I'm referring to the water, the flora, the fauna, the light, etc. Most cichlids on the market are captive bred, which means they were born and raised in tanks that probably lack anything resembling the habitat from which their ancestors came.
So what exactly does this have to do with keeping live plants? A lot, actually. Some cichlids are natural diggers/sifters (e.g., the earth eaters of South America). Others might eat or graze on plants. For this reason, having plants that aren't firmly rooted might result in a constant battle between you and your fish. For some species, plants are natural. For others, not so much. For example, most New World cichlids come from an ecosystem teaming with vegetation, so plants are the norm. The Rift lakes of Africa also have vegetation, but for the most part it's not nearly as dense throughout the habitat compared to that in which Central/SA cichlids reside. In fact, many African species inhabit plant free zones all together.
Regardless, if you want to keep plants with your cichlids, go for it. Planted takes have many benefits. Plants provide great shelter and they can also significantly diminish sight lines, which can greatly impact a tank with aggressive cichlid species. As with maintaining any closed system such as an aquarium, it's advisable that you do your homework before you make a decision.
The Cichlid Room Companion
Tropical Fish Hobbyist
American Cichlid Assoc.
African Cichlid Hub