Make no mistake, talent is (or should be) the primary criterion for selection into an international side. Once you're established in the team, then your performance as measured by statistics can certainly be used for or against you. But if performance was the primary criterion, you wouldn't need good cricketers as selectors - a statistician, heck even a computer program would serve the purpose. So what exactly is it that the selectors look for?
Now the answer will of course differ for the three disciplines of batting, bowling and fielding. So I'll try and address them one at a time. Let's take the easy one first - fielding. A talented fielder is surely one who is athletic, quick on his feet, etc. A catcher should have quick reactions, steady hands, and the movement of the hands to take catches should be smooth, not jerky. Think of a Jacques Kallis in the slips, steady as a rock. His hands always seem to end up in the right place for a slip catch. A fielder in the "ring" should be able to quickly change direction, dive, hit the stumps from 25-30 yards out, etc. No better example than Jonty Rhodes of course, or Ricky Ponting. Some of these attributes can be learned with lots of practice (like hitting the stumps), but others are natural, and thus constitute fielding talent.
On to bowling. Let's take fast bowling first. A naturally talented fast bowler would probably have to have good build (and height), and the ability to channel his muscular strength into generating pace through the air. He should have an action that lends itself to extracting bounce from the wicket as well. Now actions can be modified over time with coaching, but if it comes naturally to the bowler, chances are that he'll be more successful than one who is coached. Think of Michael Holding, or Alan Donald, or Dennis Lillee before his injury. Natural born fast bowlers, not just in physique and action, but also in the mind. Of today's crop, only Brett Lee and Dale Steyn seem to fit the bill.
What about swing bowling? Why do some bowlers seem to get more swing than others, in the same conditions? Here, it seems to be more about your action than any physical attributes. The coaches can show you how to hold the ball, angle the seam, etc. But eventually it comes down to the action that feels most natural to you. Some bowlers automatically seem to find the right arm action, cocking of the wrist, grip on the ball, etc. That's a naturally talented bowler. Think of Irfan Pathan, before it all went awry. Rhythm, and a clear head, also seem to play a significant role in swing bowling. These are natural attributes as well, and can't really be coached.
As for spinners, many different techniques have been developed over the years for the different kinds of spin. It appears to me that most of these are coachable, learnable, and thus don't constitute natural talent. To some extent, rhythm in the run-up and bowling action plays a role. But the imparting of spin doesn't appear to be a natural talent. It is usually learnt and practised over years of junior cricket. The great spinners (no better example than Shane Warne) manage to land the ball in the right areas, or flight it teasingly, or disguise the wrong 'un just a bit better than the ordinary ones. There's of course the ability to out-think the batsman - to understand his mindset, predict his intent, and bowl accordingly. But that's more of a native intelligence than bowling talent.
And now for batting. As discussed in that earlier post, batting class is hard to define and yet easy to spot. But there must be more to batting talent than class, because there are plenty of batsmen who were talented and successful, but you wouldn't call them classy. There are plenty of examples of those, like Alan Border, Javed Miandad, or Ricky Ponting. It seems to come down to instinctively making the right body movements when batting. Of course you need the ability to pick up the bowler's length and line at (or very soon after) the point of release. You need to pick the spinners out of their hand, or while the ball is in flight. But I get the feeling that these things can be coached.
What cannot be coached is the response, once you've picked the bowler and made a prediction as to how the ball will behave on its way to you. The movements of the feet - forward or back by the right amount, getting to the pitch of the ball or making room for a horizontal-bat shot, etc. - determine the batsman's success at tackling the delivery, along with things like the transfer of weight onto the correct foot (front or back) at the right time. That's what gives you the right shot selection and timing. The ability to play several different shots to the same ball is probably something that can be coached and practised. If the mind is clear, and sharp enough, the batsman can process all this information and pick a shot to play. The body movements that follow though, must come naturally and at the right time, down to fractions of a second. Of the modern greats, Tendulkar, Ponting and Lara are good examples of this. They're rarely caught on the wrong foot, making the wrong movements.
Some batsmen also appear to be better at placement than others. Think of Saurav Ganguly and Rahul Dravid, especially in ODIs, in the 1990s. Ganguly would usually be faced with a packed off-side field, and yet manage to find the gaps regularly. The young Dravid would often play these wonderful looking flicks and drives, straight at the fielders! Some batsmen just hit the ball in the right fashion, with great timing and all that. Others seem to be able to use the bat to give direction to the ball. Another example - contrast Sehwag's off-side strokes with those of his partner Gambhir. Sehwag clearly places the ball into the gaps - his strokes, his arm, foot and upper body movements are designed to give direction to the ball (except when he's in his mad hitting moods of course). Gambhir is more coached and thus more likely to find the fielders.
Aside from these physical aspects, of course there's also the mindset, the attitude, the psychology of the batsman. I'm not really considering how a batsman responds to a particular team situation - whether he plays safe when the team is under pressure, or attacks when quick runs are needed, or picks certain bowlers to go after, certain fielders to pressurize. While those attributes are important, they don't really constitute raw batting talent. It's more about cricketing intelligence, and even if it can't be made intrinsic, it can certainly be coached. So if I was a selector, I wouldn't give primacy to that aspect in picking talented cricketers.
Watching age-group or league-level cricket is great fun, because you can pretend to be a selector and look for the talented ones. Now that's the "Cricket Stalker" in me talking :)