For example, $f(x)=\sin x$ changes concavity an infinite number of times, $f(x)=x^3-x$ has two regions of concavity (changing concavity once), and $f(x)=x$ changes $0$ times.
Is there a name for this property?
The points where a function changes concavity are called "flex points" or "points of inflection."
For functions with continuous derivatives, the changes in concavity occur exactly at the local extremes of derivative. Since a polynomial of degree $k$ has at most $k-1$ local extremes, it follows that a polynomial of degree $n$ has at most $n-2$ points of inflection. Moreover, by considering the sign of a polynomial as $x\to\infty$ and as $x\to-\infty$, it is easy to check that an odd degree polynomial will have an odd number of points of inflection, and an even degree polynomial will have an even number of points of inflection, except for polynomials of degree $1$.
In particular, your $f(x)=x^3-x$ cannot change concavity twice: it has at most (and in fact, exactly) one point of inflection.
Note that this simple analysis also means that polynomials of degree $3$ change concavity exactly once, and polynomials of degree $2$ never change concavity.
I think you're looking for the property which describes the place at which a function changes concavity (point of inflection), and then, if you want, you can say the property is "$f$ has $n$ points of inflection." Generally we wouldn't give a name to a property unless we study the property in depth. And AFAIK there are no significant theorems based on the number of times a function changes concavity.
However, there are some problems being studied on real points of inflection on algebraic curves in a more general (tropical) setting. In the recent (2011) paper included below, the main theorem is:
Note that they still call the property "having $n$ inflection points"