Part of the trouble with a multi-planetary civilization is that you need to measure time in a way which is meaningful to the residents. There is a natural diurnal cycle in most species, and it's important to maintain that -- how you choose to reckon time is actually very significant when you start to think about that, because if you choose unwisely, you could end up messing up everybody's circadian rhythms. Also, you need to be able to divide time in a way which is a) not too cumbersome and b) relates well to whatever the natural unit of a useful "day" is going to be.
This problem has already been addressed in the space program. ISS crews have to have a day-night cycle, even though their "day" is about 90 minutes long (the duration of one orbit). Since there's no useful correlation to the cycles of light and darkness outside the station, they just go by what ground controllers are most comfortable with, and that means they have to periodically adjust their internal "clocks" so that they mesh with ground controllers. If there's a Soyuz taxi mission coming up, they want to be fairly close to Moscow time, for instance, though I think mostly they stay about halfway between Moscow and Houston so it's easier to slide one way or the other.
Unmanned missions to Mars have faced this problem as well, but with a different imperative -- while the ISS crew wants to keep their clocks at a cycle that will be useful for ground controllers, Mars landers have to optimize daylight. This is especially true with the modern solar-powered rovers such as Spirit and Opportunity. In order to do this, the missions don't run on Pacific Time, even though they're run from JPL in California (IIRC). They run on local Martian time. Yes, a system of reckoning time has been devised for Mars, complete with time zones. Time is measured not in days but in sols. A sol is the same thing as a day, but with Mars' rotational period -- the different name makes it easy to tell what kind of a "day" you're talking about. Teammembers have had to live on Mars time, which means that their commute shifts a bit every day, given that a Martian sol is just a little bit longer than an Earth day, and many of them wear two watches -- one is an ordinary Earth watch set to Pacific time, and the other is a special custom-made Mars watch set for their rover's time zone. The Spirit crew rise at a different time than the Opportunity crew, since their rovers are at different longitudes and thus different time zones. Interestingly, they also have a 24-hour clock -- but the hours are longer on Mars than on Earth. (I suspect the spacecraft's internal computer doesn't care about that, though, and has gone merrily on with Earth time.)
Basically, they're largely ignoring Earth time because the need for Mars time overrides it. But if you were running a civilization that had lots of activity on both Earth and Mars, I think you'd have to have Mars running on Mars time and Earth running on Earth time, with clocks and calendars designed to simplify the task of correlating the two. Just as a company with business in both India and the US will have to manage two time zones, a company with business on Earth and Mars will need to manage both time zones, even although those time zones don't progress through their calendar at the same rate. There will certainly need to be some larger efforts in standardization of interplanetary time, and I suspect there will be several such experiments before a system actually sticks.
BTW, the Earth/Mars solution is workable largely because the Mars day is pretty close in length to the Earth day. The Moon has a "day" which is about two Earth weeks long; such a system would not be helpful there, and you'd have to find some other means of subdividing time. Perhaps it would simply default back to Earth time.
Some useful links: