Martinez took over from David Moyes following his departure to Manchester United and England international Osman admits the change has been encouraging.
"We have been doing bits in training and games which is similar to last year - but also bringing in some of the manager's new ideas," he told the club's official website.
"It's been new, it's been challenging but it's been good. It's different to learn new systems.
"We are learning how the manager likes to play three at the back at the minute, to complement how we play with four at the back."
Everton now face three high-profile friendlies in the USA, with Juventus their first opponents in the early hours of Thursday morning UK time.
"This tournament has some fantastic teams and every game is going to be tough," he added.
"You want to play against top players and we are certainly going to be doing this over the next 10 days or so.
"We thoroughly enjoy being away on tour but ultimately it's about being ready for the first game of the season - we are certainly looking forward to that."