The actual equivalent to a Newtonian "slice of time" in GR is a "space-like hypersurface". It is a 3 dimensional hypersurface in 4D space-time (hence the hypersurface). It is "space-like" because any two points are connected by a "space-like" path in the metric. This means that they are independent from a causality perspective, and can form the basis of forward predictions.
If this is a global hypersurface (ie doesnt just stop somewhere, or miss out regions) then it is called a "Cauchy surface".
As remarked elsewhere there is more choice in GR involved. The simplest way to see this is to note that Newtonian theory is $E^4$ with (x,y,z,t) and once t= const is chosen at the space origin, say, the entire hypersurface is now determined.
In say Minkowski space (the nearest GR equivalent to Newtonian space) there is a choice of timelike vector to be made first. There are 4 degrees of freedom here, but there is the timelikeness restriction and the restriction on time direction. (Physically this is a choice of Inertial Frame.) This allows a foliation to form. To obtain an actual hypersurface (hyperplane in this case) we can choose one point P on that hypersurface (ie a specific time in the Frame). Now the timelike vector is a normal to that hyperplane and we can determine whether an arbitrary point (x,y,z,t) is on that hyperplane. There will be 3 degrees of freedom as expected.
GR generalises the Minkowski example in that the hypersurface might be curved and not flat, requiring an infinite number of independent points to identify it.
One further generalisation is that in Relativity we have a metric which allows null distances between different points. Thus we can have a "null hyperplane" and "null hypersurface" too, which are quasi-generalisations of the timelike slice Newtonian idea.