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We know the relativistic view: "moving clocks tick slower than stationary clocks". We also know that Einstein's Special Relativity (SR) demands: "there is no absolute movement, only relative movement". This tells us that when two clocks are moving relative to each other, we can view any one of them as the "stationary" reference and the other one as the "moving clock".
An apparent paradox arises out of this opening paragraph: if clock A is considered as the reference and clock B as moving, then B ticks slower than A. However, if B is considered as the reference and clock A as moving, then A ticks slower than B. This is a logical paradox, but it obviously arises out of the wording. It should have been: "if clock A is considered as the reference and clock B as moving, then B ticks slower than A, as observed by A. If B is considered as the reference and clock A as moving, then A ticks slower than B, as observed by B."
What does this tell us about the "real tick rate" of moving clocks? Actually nothing, except perhaps that there cannot be an absolute difference between the tick rate of clocks A and B, provided that they are moving inertially in free space. The differences are relative (or apparent) and caused by a measurement issue. What is this measurement issue?
The isotropy of the propagation of light
It originates in the second postulate of SR – Einstein's assumption that the speed of light must be constant in every direction, irrespective of the inertial movement of source or observer. From this follows the Einstein definition of simultaneity, which boils down to a convention for synchronizing static clocks inside inertial frames.
Einstein made the reasonable assumption that one can send a time-stamped light signal to any observer that is stationary relative to us. The receiver, knowing the coordinate distance from us, can then add the light travel time to that distance and set her clock accordingly. This is known as the "Einstein method" for synchronizing clocks in inertial frames. This, of course, ensures that light will propagate at c in all directions, as measured by clocks thus synchronized.
The rate of clocks
The Einstein synchronization method also ensures that we cannot measure the "real" rate of inertially moving clocks. We need two clocks (say A and C) that are separated in space to compare the rate of the moving clock (say B) to. B will however not agree that clocks A and C are synchronized and is entitled to say: "you guys can tell me nothing about the rate of my clock versus yours, because your two clocks are not correctly synchronized, IMO".
 Amazingly, it seems to be impossible to measure the relative rate of two inertially moving clocks during a single pass – you need at least two passes. To achieve that, you must take one (or both) of the clocks out of being inertial – you need a gravitational field or acceleration, both of which destroy SR's applicability.
A different view
As a point for discussion I want to put forward this view: clocks in relative inertial motion all tick at some universal rate, but we have no means of detecting this. What we can detect is the relative, coordinate dependent rates of clocks, which depend on a convention for synchronization of clocks. It is a very good convention, it seems, because SR predicts exactly what we actually measure.
The situation is best described my means of a Loedel diagram[1], as pictured here. Objects static in each frame (A or B) have wordlines progressing at the same rate up the respective time axes. The dotted red and blue lines of simultaneity do however indicate how each view the others clock as "running slow". When the blue observer's clock reads t=4.25, she observes the red clock as reading only t'=4.0 and vice-versa.
Jorrie
[1] The Loedel diagram is due to E. Loedel (Geometric Representation of the Lorentz Transformation. Am. J. Phys. 25: 327, May, 1957). It is just a special case of a Minkowski diagram, with two inertial frames in symmetrical motion in opposite directions relative to some imaginary "neutral" inertial frame. When precisely two inertial frames are required, the Loedel diagram is perhaps the easiest (rigorous) representation of SR. You can read more on Loedel diagrams on my website.
Edit: the Resolution
The view presented above says: "clocks in relative inertial motion all tick at some universal rate, but we have no means of detecting this."
This sounds very compelling (especially if we have no way of checking it!), but there are some clues that it may not be the case. We have no means of directly comparing the tick rates of two clocks in relative inertial motion, as they will have to be brought together at the beginning of the test and then again some time later. This means that at least one of them must be accelerated at least once; this renders that clock to be no longer purely inertial.
 However, we can use three purely inertial clocks to perform such a test. In the usual "twin paradox" thought experiment, we substitute (at the turn-around point) the "away clock" with another clock, flying towards home. Now there is no acceleration required and two clocks at a time can be directly compared (and synchronized) as they fly past each other.
The left-hand dates are as recorded by the home clock, the blue line is for the away clock and the red line is for the substitute away-clock. On New Year's Day 2007, the away twin zips past the home twin and sets her clock to read the same date. When her clock reads New Year 2011, another traveler, carrying an identical clock, passes her in the opposite direction, but at the same speed (v=0.6c) relative to home. At the 2011 passing, her "substitute clock" is set to that date as well.
When the substitute away twin passes home (home time 2017), Einstein's special theory of relativity tells us that her clock will read 2015. This means that the combined elapsed time of the away twin and her substitute was less than the elapsed time of the home twin, despite the fact that all three of them were purely inertial during the whole experiment.
Does this tell us that there are some inertial frames that are privileged - inertial frames in which clocks run slower (in an absolute sense) than clocks in some other inertial frames? Not quite. We have no absolute way of knowing what the time reading on the home clock was at the moment that the two away clocks pass each other. The fact that we draw it symmetrically does not guarantee that it must be such a symmetrical case.
What we can say is that on the two away clocks the average elapsed time (of the test) is less than the elapsed time on the home clock, because we could measure it absolutely. So what causes this difference? The clocks followed different spacetime paths - somewhere in their respective pasts they had to be accelerated differently in order to move as they did. This change in space versus time orientation apparently causes clocks to run at different rates.
Clear as mud?
Edit (Nov 16, 2009): In order to clear the 'mud' a little, the final statement above:
"... somewhere in their respective pasts they had to be accelerated differently in order to move as they did. This change in space versus time orientation apparently causes clocks to run at different rates"
is not quite mainstream thinking. There is really no way to observe that. It is easier to view the clocks to run at the same rate, but that the one(s) following the longest spatial path record(s) the lesser time differential, in other words, they follow the shorter temporal path. This comes directly from the Lorentz transformations, which have their roots in the invariant (time-like) spacetime interval:
ds2 = dt2 - dx2 - dy2 - dz2
The more distance covered between two events, the less time covered...
-J
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Re: Do moving clocks really 'tick slower'?
