I've tried Peter's UI implementation, and I've had trouble as well, so I built my own interface for Date and Time picking for Foundation. It's a little fresh, but my company has been using it on a production site and it's holding up.
https://github.com/jockmac22/foundation_calendar_date_time_picker
I built it around unobtrusive javascript, so its easy to implement through data- attributes in the input field.
I have an SCSS file you can include in your foundation SASS build to incorporate the required CSS directly into your final CSS file. This way you can customize the coloring with your existing foundation settings, and make any other changes easily inside the SASS framework.
It supports custom date, time and field value formats so you can have a user friendly display of the date and time, with an ISO standard (or any other format) as the final value stored in the field. I used the Ruby strftime formatting options as the basis for the date formats.
Let me know your thoughts if you decide to try it.