Dodds, who has been with Honda since 2002 and previously worked for Volvo, will take up the role on 1 April.
Thompson has left to head Motorola's European marketing after 13 years with the car marque (Marketing, 16 February).
Dodds is responsible for Honda's all-terrain vehicles, speedboats, engines and generators - a role Thompson held before becoming head of marketing for cars.
During his time at Honda, Thompson won critical acclaim for the 'Cog' campaign, which was created by Wieden & Kennedy in 2003. He also oversaw the marque's recent 'Impossible dream' and 'Choir' work.
Thompson took over from Chris Brown as head of marketing in 2002. He left shortly after Ewan Ramsey's return to Honda as general manager of cars after a period working at DaimlerChrysler.