How to handle newly discovered problems in second and third round reviews
(1) The comments of other reviewers should not bother you; often reviewers will be from mutually exclusive fields, so as to provide better representation. You should stand by your professional opinion irrespective of the others.
(2) I firmly believe that you should spend a lot of time on first and second review. If you are finding new and significant errors beyond these rounds, it means you didn't do your job rigorously before.
EDIT: Several comments highlight the valid point that sometimes revisions are required just to make the paper readable. My position on this is that if more than 2 'readability-reviews' are required, the authors are either unable to use the language adequately (and should be directed to a suitable resource- a reviewer is meant to check technical content, not be a style coach), or are unwilling to make these suggestions; in which case there is no point asking once again.
Comments/questions regarding your previous objections are ok, but entirely new objections in third or subsequent rounds leave a bitter taste and lead to a lot if time being wasted.
This has also happened to me, so I appreciate this question. I think this situation occurs, in large part, because a totally sloppy first draft is sent out for review rather than desk rejected. So in some sense this is not your fault.
In the end, I gave a review including all the issues I had identified and then asked the editor for feedback regarding my (second and third round) reviews. He explained that he was pleased with my approach, although in the end he ended up telling the authors that some of my comments from the third round were optional.
The feedback I received makes me believe that a reviewer should continue to point out issues in the second and third rounds, even if they are new. Then alert the editor to the fact that you are finding new issues (perhaps note that it is due to the substantial rewriting). Final decisions are up to the editor and usually they are happy to be able to consider the points, even if (in the end) they tell the author that some points can be ignored at the later stage.
I appreciate the fact that you are asking this questions with sensitivity to the authors' feelings. Although we should be professional and objective in doing our reviews, we should also respond in ways that take the authors' human feelings into account.
In this case, I think you should indeed present the new concerns you have at each round, especially if they were hidden earlier do to poor writing. However, when you do so, I recommend that you send a note to the editor explaining your concern that it might be unfair to the authors to bring up new major issues so late, and so ask the editor to use their discretion in making the recommendation. That way, you fulfill your responsibility to objectively identify the problems with the article, but you absolve your conscience of any possible guilt for giving the authors an unfair experience--it is the editor's responsibility to make the decision, not yours.