As I understand it, pretty much everybody uses the "Babylonian/sexagesimal" time format: 12/24 hours in a day, 60 minutes in an hour and 60 seconds in a minute.

Have there been alternative systems to accurately tell time? Or maybe there are still some in existence?

By "accurately" I mean that one can tell an exact time, i.e. not the "some time during the morning"-type. Also, let's exclude inventions that were never properly used over a longer period, e.g. decimal time during the French Revolution.