Rosa, who had denied the charges made against him in July, runs Italy-based company Rosa & Associati.
Kenya, renowned for its distance runners, has faced frequent allegations of doping with some 40 cases reported in the past four years.
"The case is withdrawn," Bernard Ochoi, Nairobi's principal magistrate, said in a statement delivered in court before Rosa's trial began.
The statement was sent to Reuters by Rosa's lawyer James Nyiha.
"My client should not have been charged in the first place. The state did not have an iota of evidence against my client," Nyiha said.
Kenya claimed its biggest ever haul of medals at an Olympic Games in Rio de Janeiro this year, but doping allegations and organisational problems plagued preparations for the event.
(Reporting by Humphrey Malalo,; Writing by Edmund Blair, editing by Ed Osmond)
