If you are just using dates in narration, as opposed to in dialog, you could just use Gregorian dates. When an American or European writes a history book today, they routinely use the Gregorian calendar even if that's not the calendar used by the people they're talking about. This only matters if the date itself is significant to the subjects.
If the people on the other planets are colonists from Earth and not aliens native to that planet, it's quite possible that they WOULD use an Earth-based calendar. Especially if they're in regular contact with Earth people. Sure, the Gregorian calendar does not match the orbital period of Mars or Pluto, but the convenience of using the same calendar on all the planets in the solar system would likely outweigh any advantage of having the calendar match the orbital period. I think this is what most science fiction stories do: They just have the characters express time in "standard Earth years" and leave it at that. It's not implausible, because the same reasons that make it simpler for the reader would make it simpler for the people involved.
Of course if you're talking about aliens, I wouldn't expect aliens to use the Gregorian calendar. That would be a rather jarring disconnect.
So failing that, if for one reason or another you must have these people have their own calendar, I don't see how there is any way out of explaining it, at least to some extent. You can't just say that one event took place on the Third of Foo and another took place on the Lesser Mordac of Hegenev and expect the reader to somehow know which came first and how far apart they are.
Unless you want the workings of the calendar to be a major element of the story, even a simple calendar would be a pain to the reader. Like if you say that the Martian calendar has 20 months and here are their names, and the months are 33 days each except for the 2nd, 4th, 7th, 9th, 11th, 13th, 16th, and 17th, and the 19th on leap years, and leap occur two out of every five years, etc. (I believe that adds up to the right number of days, if I made a mistake, not really the point), that's about as simple as you're going to get, and it's an awful lot for a reader to understand and remember just so he can grasp when things are happening.
I think the easiest way out is just to translate the dates. As I said, if it's in narration, you can simply say, "this happened on August 3". What the date is on somebody else's calendar doesn't matter. If it's necessary for a character to say the date, you could translate. Like:
"We must do this by the High Mordac of Hegenev," Franslac said.
George did a quick translation in his head: That would be August 3.
Of course if you're doing this all the time, that could get tedious for the reader. That's why, I think, most writers avoid the problem.
Addendum 3 years later
When we give alien dialog in a book, we routinely give it in English, even though it would seem somewhat unlikely that aliens from another star system would speak English. We have the aliens give distances in light-years, even though they would surely not use this particular unit. Etc. The reader understands that you are translating. So translating dates also would not be implausible.
Where it WOULD get implausible is:
(a) Anything that assumes that dates "line up". Like, if a crucial plot point hinges on an event happening at the stroke of midnight on July 8, I wouldn't expect that to be the same time at Star Fleet Headquarters on Earth as it is on the capital of the alien planet. That doesn't work with time zones here on Earth. Similarly, we wouldn't expect the length of a day or a year to be the same.
(b) I wouldn't expect aliens to attach cultural significance to the same days we do. If in a book an alien says, "The invasion will begin on November 12", I'd accept that as a translation. But if an alien says, "We're delaying the invasion until after the new year", that seems to say that they start their year on January 1 like we do. "The invasion will begin the day after Christmas" -- unless you've given some reason why the aliens celebrate Christmas or at least are aware of it, this would be very jarring.