A petition accusing Disney of "colonialism and robbery" for trademarking a phrase used in the film The Lion King has attracted more than 30,000 signatures.
"Hakuna matata" means "no problem" or "no worries" in the Swahili language, which is spoken across East Africa.
Disney first applied to trademark the catchphrase in 1994 - the same year it released The Lion King animation.
The row has arisen now amid hype over its live-action remake.
Petition organiser Shelton Mpala hopes he can persuade the US company to abandon its "hakuna matata" trademark.
"A lot of Swahili speakers have been utterly shocked, they had no idea this was happening," Mr Mpala told the BBC.
"Growing up in Zimbabwe, I always had an understanding that a culture's language was its richness."
Disney has not responded to the BBC's request for comment.
The Lion King franchise also includes a stage musical, merchandise such as T-shirts and toys, and computer games, plus several film sequels and TV spin-offs.
Mr Mpala says he is not a Swahili speaker but he believes the trademark is merely the latest example of Africa being "exploited in some shape, fashion or form".
He is not alone.
Some Kenyan newspapers have also accused Disney of stealing from Kenyan culture by claiming ownership of the phrase, and the current debate appears to have started with this article on the Business Daily Africa news site.