Hick will act as assistant coach to Justin Langer, who will be in charge for the series while Darren Lehmann has a break, while Griffith replaces Craig McDermott who has stepped down.
Head coach Lehmann said both would be a great help to Langer, the former Australia opener and now highly regarded coach of Western Australia.
Hick has worked with Australia's National Cricket Centre for the past three years, focusing on young players coming through the academy, while Griffith is the WA bowling coach.
Australia begin the tri-series, which also features South Africa, in Guyana on June 6.
The round-robin tournament sees teams play three matches against each opponent, with the top two advancing to the final on June 26 in Barbados.