The 28-year-old defender thinks clubs will use the added incentive of Champions League qualification to compete, after newco Rangers were denied entry into Scotland's top-flight.
And although Zaliukas says that the league won't be the same without Rangers, he now wants his own club to be amongst the challengers.
"I think it is probably going to be harder now because all teams will want to achieve second place and it is going to be a hard season," Zaliukas told reporters.
"It [Rangers] was a good club, a brilliant stadium and great atmosphere to go there.
"Obviously fighting for honours is our target, but it depends how we negotiate the season.
"Before the start of the season, our target is to get the maximum, compete with Celtic and do well in both cups."
Hearts ended last season in emphatic fashion, hammering rivals Hibernian 5-1 in the Scottish Cup final, and whilst Zaliukas won't be forgetting the triumph in a hurry, the Lithuanian defender wants Hearts to retain the trophy.
"Every time anyone can touch it, they should do it. We want to do well again this year and have the trophy again," he added.
"Last year's win is in the past already, but it has taken a place inside our hearts for the rest of our lives.
"Now it is time to start again and try and achieve this again."
Hearts go into the upcoming season under new management, with former Raith Rovers boss John McGlynn at the helm, but a lack of funds has left the 50-year-old looking to the club's youth players.
Zaliukas believes this should be looked upon as a positive, and insists that all is well at the club ahead of the new campaign.
"Pre-season is going quite well," he said.
"The training is good and all the youngsters are performing well and putting pressure on us older ones to work harder.
"I think everyone is enjoying working under the new manager, all the training is good and we are ready to go."