The Spaniard enjoyed an impressive debut campaign at Goodison Park, guiding the club to fifth with a free-flowing and entertaining brand of football.
While Everton missed out on UEFA Champions League football in the final weeks of 2013-14, Martinez's men will compete in the Europa League next term.
And Osman feels the current style at Everton suits both him and his team-mates - even if they had initial reservations about Martinez's approach.
"At first we thought we were too expansive and opening up left us vulnerable to being done on the counter-attack," Osman told the club's official website.
"But we quickly learned how to deal with that and it has been positive.
"It is a style of play that is possession-based and a player like me is going to prefer being on the ball than having to chase after it.
"We all embraced it from the start. We are a good bunch of lads, an open bunch that are not set in our ways by any means. We were very open to what the manager wanted to do."
The form of FIFA World Cup-bound pair Ross Barkley and Romelu Lukaku has been integral to Everton's season along with that of fellow youngster and Barcelona loanee Gerard Deulofeu.
Osman feels the trio have coped well with the demands of a full Premier League season, describing it as an "incredible feat".
"It's not easy for a young lad to come in and show what he is capable of over a whole season," Osman continued.
"It's easy to do it for a game or two but then you might go off the boil.
"But these lads have pretty much done it for the whole season and for lads so young, that’s an incredible feat."
