Very similar, but there are some significant differences. First, C# supports constructor chaining. This means one constructor can call another:

class Person
{
public Person( string name, int age ) { ... }
public Person( string name ) : this( name, 0 ) {}
public Person() : this( "", 0 ) {}
}
Another difference is that virtual method calls within a constructor are routed to the most derived implementation - see Can I Call a virtual method from a constructor.

Error handling is also somewhat different. If an exception occurs during construction of a C# object, the destuctor (finalizer) will still be called. This is unlike C++ where the destructor is not called if construction is not completed. (Thanks to Jon Jagger for pointing this out.)

Finally, C# has static constructors. The static constructor for a class runs before the first instance of the class is created.

Also note that (like C++) some C# developers prefer the factory method pattern over constructors.