When a set has only finitely many elements, you can display it by listing its members: the set of digits base ten, for instance, is $\{0,1,2,3,4,5,6,7,8,9\}$. When a set has infinitely many members, you can’t list them all. If they can be arranged in a sequence that has an obvious pattern, you can suggest the set by listing enough of its members to make the pattern clear to the reader: $\mathbb{Z} = \{\dots,-2,-1,0,1,2,\dots\}$. In many cases just about every reader will make the same interpretation. For example, I’d expect practically everyone to interpret $\{2,4,6,8,\dots \}$ as the set of positive even integers. But even in simple cases like this you can’t be sure that everyone’s pattern matching sense will agree with yours, so if you really want to specify this set precisely, you have to write it in a way that includes a precise specification of what qualifies something to be a member of it: $\{n \in \mathbb{Z^+}: 2|n \}$, $\{2n:n \in \mathbb{Z^+}\}$, $\{n:n \mbox{ is a positive even integer}\}$, etc.
Moreover, the ‘suggestive listing’ technique is possible only with countably infinite sets, and not even all of those: it has to be a set that can be arranged in a ‘nice’ sequential order, one that a reader can reliably identify with the set in question. The set $\mathbb{Q}$ of rational numbers would be very difficult to indicate in this way with any real assurance that virtually all readers would correctly identify it.
In your problem there is no hope of such a listing, because the set of real numbers is uncountable, and therefore your set $A\times B$ is also uncountable. This guarantees that there is no way to suggest its membership by listing a few members and expecting the reader to spot a pattern that says ‘These are the real numbers’. You have no choice but to specify the set precisely. There are many slightly different ways to do so; Qiaochu gave one in the comments. A slightly different one is $A\times B =$ $\{\langle x,n \rangle:x \in \mathbb{R} \mbox{ and }n \in \{1,2,3\}\}$. A very short one: $\{\langle a,b \rangle:a\in A \& b\in B\}$. Yet another, much wordier: $\{\langle a,b \rangle:a\mbox{ is a real number and }b\mbox{ is }1,2,\mbox{ or }3\}$. (The short one probably isn’t a good choice if you really want to give the reader an immediate clear indication of what the elements of this Cartesian product actually are.)