The Irish finished the championship in third place after being defeated 30-9 by England at Twickenham in the worst defeat of coach Declan Kidney's reign.
Next in store is a daunting three-Test series against the world champion All Blacks, a side they have never beaten.
"We're really going to have to pick ourselves up if we're to avoid being made a joke of in New Zealand," Bowe said.
"We're very disappointed and that performance against England was a complete let-down. We know we're better than that.
"We'll need to have a look at ourselves because it's not good enough.
"That defeat has left a very sour taste in our mouths and it was a very quiet changing room."
Bowe insists Ireland should have been on course for the title when they arrived at Twickenham.
Instead, they were flattened by a vastly superior England team that used an overwhelmingly dominant scrum to pound the visitors into submission.
"Consistency is the problem that is letting us down," Ospreys wing Bowe said.
"We have shown in the past that we are a top team and against England we should have been playing for a championship or a Grand Slam.
"We were ahead against Wales despite not playing that well and we were ahead against France until a bad 20-minute spell in the second half cost us the campaign.
"I hate saying we're not far off but at times we've shown that we're better than these teams and are good enough to compete.
"I don't know if it's in our heads, we've just switched off at times. It's something we must look at."
Ireland are waiting to see if citing officer Alberto Recaldini takes action over an accusation by flanker Stephen Ferris that he was bitten during the first half.
Ferris protested to referee Nigel Owens, who claimed he did not see the incident, and then received treatment.
Neither coach Declan Kidney nor captain Rory Best would elaborate on the incident after the game.