tl,dr: This is really just a verbous synthesis of what has already been said by alanf and CuriousOne with a more basic experiment and theory explanation approach and a smattering of my own limited knowledge.
The upshot is the same: trajectories make little sense in nonrelativistic QM and if you factor in QFT, you can't even speak about particles.
The problem here is that our intuition is ill-defined when it comes to quantum mechanics. In particular, a "particle" is ill-defined. A second problem is that there are two sides to this question: theory and experiments - and another problem is that quantum experiments are much more difficult on a fundamental level.
Experimental Trajectories
Let's start with experiments (one should always start there, right?) and suppose we know what a "particle" is. Now, for classical physics, we can all agree that a body has a nice trajectory like a football shot on a goal. When you ask people whether quantum particles have trajectories, they will point to bubble chamber trajectories and violà, there are clear particle trajectories. Or they might point you to particle accelerators, where particle trajectories are visualised. As mentioned in another answer, the trouble is that the quantum particles here have very high energy. If the amount of energy involved is very high, the particle, albeit tiny, will behave classically where we are concerned about.
Thus, in order to see the "true" quantum world, this won't do. You'll have to think of low energy particles - maybe electrons in an atom or such. When you want to look at the "trajectory" of such a particle, you'll already be in trouble: you can't just shine light at it and watch it move - any photon with a moderately high energy will just ionize the atom and completely change whatever "trajectory" you wanted to look at. And even if you don't ionize the atom, you'll still heavily influence the path. If you want to "watch" a photon, you could use photo-detectors to measure the exact place where the photon is - but afterwards, it'll be destroyed, so that's also no good to measuring a trajectory.
First lesson: What you can do is already limited to one measurement of position or momentum. Everything after that just won't make much sense any more.
From Experiment to Theory
Now you might be clever and think: I can just set up an experiment, where I sent an electron time and time again and if I keep everything else fixed, every electron will have the exact same trajectory (this is what we expect from classical physics) and then I can just measure the trajectory by measuring speed and/or position at different places.
This is where the Heisenberg uncertainty jumps in and tells you that the result will never be a single trajectory. However hard you try, there will be a fundamental distribution of the results. If you measure "one" trajectory (meaning you prepare an electron, measure its position after time $t_1$, then reprepare, measure after $t_2$ and so on), you'd get an eratic behaviour. If you repeat the measurements at every point, you'd get a distribution of results. This is one way to say "particles don't have a trajectory".
Second Lesson: Repeating an experiment won't help - the outcome will be a probabilistic trajectory, not something you'd normally call a "trajectory".
However, what you can observe is how motion is built up: The probability of where to find the particle changes over time as does the speed. If you think you are sending a particle along some linear trajectory, what you will observe if done right is that the particle will "probably" move along this trajectory.
Nonrelativistic Quantum Mechanics
You can even go further: If you look into quantum mechanics (which can perfectly describe anything I have just talked about), in the scenario of an electron moving along some path* we have Ehrenfest's theorem, which tells us that the expectation value of position and momentum behave like a classical particle in classical mechanics: the expectation value has a classical trajectory! Note however that since the probability distributions are not very sharp, we cannot say the same for a single particle.
In nonrelativistic quantum mechanics, people tend to interpret this (and some other results together) as meaning that no particle really has a well-defined position and/or momentum. For large momenta (i.e. large energies) or large particles this doesn't really matter because the probability distributions become very sharp, but for our electron at low speed it does. This doesn't mean that the position is just a bit uncertain, it means that asking for a position of a particle when you don't measure it doesn't make sense. Note that this is a theoretical statement. If you don't measure, you can't really say anything, but using measurements you can rule out simple theories where a position and momentum are well-defined.
Third lesson: Mostly, people interpret quantum theory to imply that drawing the movement of a particle doesn't make sense. The particle moves neither along an eratic, nor a straight trajectory and it also doesn't magically disappear at a point and reappear at another. The question "How does the movement look like" just doesn't make sense (and you will always fail when you answer it experimentally).
Can we get around this?
Yes. Bohmian mechanics, a different yet equivalent interpretation of nonrelativistic quantum mechanics assigns each particle a position and a trajectory "pilot wave". This trajectory uses hidden variables (i.e. variables which we have no experimental access to). Note that the predictions of Bohmian mechanics and canonical quantum mechanics are the same, it is (so far) only a reinterpretation of the theory.
However, let me warn you: Bohmian mechanics has many problems of its own, not least that (as of now) it cannot really make sense out of QFT.
More theory and philosophy
I have always talked about particles and nonrelativistic quantum field theory. As pointed out by other answers here, the story gets even more complicated with QFT (from the classical intuition - it actually gets simpler in a certain way).
You seem to believe that there are particles and a massive particle must of course have a place where you can find it. There is a rest frame for the particle, so I could just go and have a look at it. Sadly, that's not quite true. The fundamental objects in quantum field theory are not particles, they are fields and their excitations. The connection to experiments is the rule that a quantity in an experiment corresponds to a self-adjoint operator and the values that the experiment will give are elements of the spectrum of that operator. Quantum fields have a ground state and every other possible state is an excited state. Very often, the results of the measurement are discrete ("quanta") and therefore, so are the excitations. Often, people therefore call these excitations "particles". The excitation of the electromagnetic field is a "photon" and there are quantum fields where the excitation is an "electron". However, we have a problem: These "particles" are not really what we think of as particles. They might not be localised - i.e. they really don't need to have a "position", i.e. if you look at the values of the field everywhere, it might be that the excitation is very spread out and there is no place where you could really say "Here is the particle". They might be somewhat localised and look like "lumps" or they might not be. In a sense, the orbital of an atom is a visualisation of the electron-excitation. And looking at such an orbital, I wouldn't really call it a particle, because - well for one, because it doesn't have a well-defined position.
And this is a basic problem of quantum field theory: Excitations are rarely localised. Talking about particles as localised objects with definite properties will only ever (partially) make sense in the absence of interaction. There are mathematical results to back this up (see this entry in the Stanford encyclopedia to learn more about the ontology of QFT).
Last lesson: In other words: Since we can't even define particles (localised objects), we don't even have to care how they move from point to point or how their trajectory looks like - none of these questions has meaning. The only thing that we can talk about is the probability density corresponding to observables with respect to quantum fields. We can ask how they evolve (this is answered by QFT) and we can have a look at asymptotic behaviour and how classicality emerges.
Last but not least: In the LHC experiments alluded to earlier, we can now say a bit more: After the collision, we have a bunch of excitations with a very high energy. They are already in the asymptotic regime: the detectors interact only weakly with those particles and therefore they behave very much like particles. Since in addition their energy is quite high, their probability densities are very sharp so that you really do behave like a particle trajectory. The interesting part of course is the collision - where the interaction is certainly not small. At that point, we cannot make sense of the words "particle".
*(see how intuition dictates how I have to write this sentence? I'm trying to explain that the sentence doesn't quite make sense, yet have to write it down to explain the scenario...)
