one of the first things we're told in many statistics texts is that the sampling distribution of the mean is normally distributed
Well, no, we're usually told something different to that.
[If you can find a reference that actually says what you said, I can show why they're wrong easily enough. But most texts don't say that.]
In practice, when is the sampling distribution of the mean actually normal? For iid random variables, I think the sample mean is only going to be actually normal when the individual components are .
How often will the sampling distribution of the mean not be normally distributed?
Well, in truth, it's probably never actually normally distributed. However, often the distribution of sample means will be very well approximated by a normal.
The obvious next question is "How often will it be 'close enough'?".
That depends on
(i) your circumstances (some applications - almost never; other applications, quite often);
(ii) your sample size (with really large samples you'll see it more often than if your typical sample sizes are small); and
(iii) on your tolerance for deviation from normality (i.e. how close is close for you? -- we can't tell you that).