If it is not already a Date, parse it into a Date. The date format is arbitrary as long as you can construct an appropriate SimpleDateFormat to represent it.
After you have a Date, you can use Date.getTime() to retrieve the millisecond value.
For the example you have shown, if you have a string:
String datestr = "2014-14-03 01:39:00";
Then the matching SimpleDateFormat would be:
DateFormat format = new SimpleDateFormat("yyyy-dd-MM HH:mm:ss");
And conversion would be:
long millis = format.parse(datestr).getTime();
It's no problem to use Date for this, as the constructors and getTime() are still some of the few remaining non-deprecated components.
Edit: I see that you have edited your question to include the use of Date. The constructor you are using is deprecated, and is also not very flexible wrt. input (you have to have the date components already parsed to use it). A SimpleDateFormat provides a non-deprecated way to convert arbitrary strings to dates.