I'd make people use the following calendar:
It divides up the 365.25 (rounded off) day year into four quarters where each of the three months in each quarter have 31, 30, and 30 days. Each quarter starts on a Sunday and ends on a Saturday. You can use this calendar every year. Every date always falls on the same day of the week each year (my birthday will always be on a Tuesday).
Since 4 months of 31 days (124) plus 8 months of 30 days (240) adds up to 364 days, we need another day. This calendar solves the problem by creating a Worldsday (W) after December 31 and before January 1. It's a global holiday and is not assigned a day of the week. Technically, a day like this is called an intercalary day (extra days inserted to align the calendar).
Because there's 1/4 day every year we have to worry about too, they insert another Worldsday after June 30 and before July 1 every 4 years (Leap Years) - unless they're centennial years (those ending in -00) not evenly divisible by 400 (that's what we do with our present calendar too, by the way).
I think it's pretty cool. It would actually save society tremendous amounts of time and money. As an example, every two years, when we update our college catalog, committees of people meet for hours to develop the academic calendar for the next two years - when classes start, when all the holidays are (if a holiday falls on a Tuesday, we need to make up that Tuesday somewhere else), when final exams are, etc. You also don't have to print new calendars every year. Imagine the savings for financial institutions in calculating quarterly reports. With this calendar, you make up a schedule once and it's good for perpetuity.
The only glitch in all of this are some religious holidays (e.g. Yom Kippur, Passover, Good Friday, Easter) which are based on the ancient Hebrew lunar calendar and have the holidays bouncing all over the place. That's the main objection - a literal reading of the Judeo-Christian Bible holds that the 7th day (Friday? Saturday? Sunday?) is the Sabbath and needs to be kept holy. The intercalary Worldsday disrupts that cycle and I guess some think it will make God mad.
Why don't we use this great calendar? Who knows, people have been advocating its adoption since 1930.
Read more about it at http://www.theworldcalendar.org/.