The 20-year-old Switzerland Under-21 international underwent a medical before putting pen to paper on a multi-year contract, completing a permanent move to Liguria.
Meichtry arrives on the back of a remarkable season, having been one of the standout performers as Thun completed an extraordinary run to the Swiss title.
The newly promoted club topped the table in a campaign few had predicted, with the young midfielder a regular fixture throughout.
He made 35 league appearances during the title-winning season and contributed eight goals from midfield, numbers that mark him out as one of the most productive young players in Swiss football over the past year.
That form has now earned him a move into Serie A, where Genoa will hope his energy and eye for goal can add fresh legs to their midfield options.
The move represents a significant step up for the young Swiss international, trading a newly crowned domestic champion for a place in one of European football's most storied top flights.
Genoa moved quickly to get the deal done, with the medical and signature completed on the same day, underlining both the club's confidence in the move and their eagerness to have their new midfielder in place as preparations for the new season continue.