Snítil made his international curling debut at the 2003 European Curling Championships, having coached the Czech men's team the previous season. Martin played second on the Czech team, skipped by his brother Jiří, and they finished 15th. The team returned to the European Championships in 2006, this time with Martin playing third on the team. At the 2006 European Curling Championships, they finished 11th. They improved the following season; at the 2007 European Curling Championships, they finish in 8th place, qualifying themselves for the 2008 World Men's Curling Championship- the first Czech team to qualify for the Worlds. At the 2008 Worlds, they placed in last place. The 2008-09 season was slightly better. Martin and his brother played on the Czech team at the 2008 European Mixed Curling Championship, where they won a silver medal, after losing to Germany in the final. At the 2008 European Curling Championships, they improved to a 7th-place finish, and at the 2009 Ford World Men's Curling Championship, they improved to an 11th place showing. Martin had played 2nd on the team at the Euros and third at Worlds.