The case for 3D time is very simple: space is based on the measurement of distance and time is based on the measurement of duration. As the distances between places cover three dimensions of space, so the durations between events cover three dimensions of time. As distance may be measured going from or toward a space-time point, so duration may be measured going from or toward a point with time-space.
In everyday life we speak of how far away a place is as a length of space or a length of time for a mode of travel. For example:
Can this informal way of speaking be formalized in the language of physics? Yes.
The difference between space and time is that distance is measured asynchronously, whereas duration is measured synchronously. One must synchronize a motion with a standard motion – called a clock – in order to measure time. A clock measures one dimension of time in one direction (or without regard to direction). A rod or rule is used asynchronously to measure a motion before, during, or after the fact – in one direction.
The context requires a mode of travel in which there is a modal rate of travel that can be used to convert space and time into one another. This is commonly done in physics with the speed of light. Length can be defined in terms of the distance traveled for a specified time at the speed of light (and the standard meter is defined that way, see here). Time can be defined in terms of the travel time for a specified distance at the speed of light. The independence of this modal rate from any particular speed of travel leads to relativity theory.
Direction is determined by the relation of a motion to a spatio-temporal point, which is either a point from which the motion comes or toward which the motion is going. For example, magnetic north is a point on earth toward which compasses point. This point moves slowly, so for accuracy surveyors must specify the year a measurement was taken.
As space has direction in three dimensions, so does time. For more about 3D time, there is a list of posts here.