Well, that's a difficult subject for me to go into myself in terms of maximum verifiable confidence, mostly because the correlations as far as the LDS are concerned would have to do with things I'm not privy to witness due to lack of membership and unwillingness to obtain an unsanctioned recommend to find out myself. I'd rather chalk up any possible similarities to the social phenomenon of two organizational units having rubbed up against each other at some point (which they did) and just-so-happening to leave a bit of a mark from the friction. It's something pretty common in the Christian denominations from what I know, and not at all unnatural in terms of social influence or attitude migration-- it could start a few converts or mutual members passing ideas, and wind up being a lasting memetic ripple in the pond of consciousness
I have the opposite verification problem, since I'm not a Mason (obviously, since I'm female

), but am LDS. It's my understanding (which I hope Bob will correct if I misstate anything) that Masonic ritual represents part of an ancient ceremony that was restored in part due to Joseph Smith's contact with Masonry (he and several other early members of the LDS church were Masons). That restored ceremony is now our Temple endowment.
I of course cannot confirm very much of this firsthand, lacking first-hand knowledge of Masonic ritual and having made covenants to not go into detail regarding Temple ordinances outside the Temple.
I've heard of it being claimed that the Freemasons date back to actual construction workers who were involved in building and maintaining Solomon's Temple; that because it was expedient in allowing them to go about their jobs, they were allowed to be present at ceremonies and rituals in which they would otherwise not have been involved, but were forbidden to discuss what they saw except among themselves. The story has it that after the Temple was destroyed, this group took it upon themselves to preserve what they could of these ceremonies; holding fast; of course, to their obligation to keep these within their group and not disclose them to anyone outside of their group.
It would stand to reason, if this claim were true, that the Freemasons of today would have ceremonies that bore some recognizable similarity to the ceremonies that actually took place in Solomon's Temple.
As a Mormon, it would not be any stretch, then, to suppose, even if no contact had ever occurred between Joseph Smith or any other church leaders, and the Freemasons, that our ceremonies may included restored forms of the original ceremonies that took place in Solomon's Temple, and would therefore bear a recognizable resemblance to the ceremonies that the Freemasons have, derived from that source.
I do not know the origin of this claim, or how much credibility to grant it. Assuming any similarity between the Masonic ceremonies and our own, this claim, if itself plausible, would certainly provide a plausible explanation for that similarity.
Now, as it happens, Joseph Smith was a Freemason. There are conflicting accounts as to just when he joined the Freemasons, relative to when he revealed the ceremonies that take place in our Temples.
In any event, I don't see how any account, accepted as true, of the connections between Freemasonry and early Mormonism creates any problem for us. Clearly, we believe that the ceremonies in which we participate in in our Temples are given to us by God, and are taking place in the context in which God intends them to take place. If anything was copied from the Freemasons, then I find no trouble in believing that the Freemasons were perhaps used by God as an instrument by which some bits of ancient knowledge and ceremony might be preserved until the time came to restore them to their proper form and context.
Of course, all this assumes that there really is some recognizable similarity between the ceremonies of these two different organizations. I think the only ones who could know this would be those who are both Freemasons and who are also Mormons who have been through the Temple. As an example of the latter, I am under sacred obligations to hold what I have experienced sacred and not disclose details thereof to outsiders. I understand Freemasons to be under similar obligations with regard to their ceremonies. Anyone from either group who purports to be disclosing such details could not be assumed to be a credible source of that which he claims to be disclosing — either he is violating sacred vows and covenants, and thus proving himself to be untrustworthy and dishonest; or else he is simply lying about the details that he claims to be disclosing.
I do know that there are many sites on the web that purport to reveal details of both groups' ceremonies. I cannot comment at all on the accuracy of the accounts of Masonic ceremonies; regarding Mormon ceremonies, I can say that there is considerably more falsehood than truth to be found on sites that purport to be disclosing our details. I won't say anything about what out there is specifically false, and what is true, other than to say that in general, the more bizarre it seems to you, the more likely it is to be false.