Consider a spinning wheel, which is held up by one end of it's axis like this: 
To explain why the change of angular momentum is directed as shown in the figure above, one usually says that there is an applied torque $\vec{\tau} = \vec{r} \times \vec{F}$, where $\vec{r}$ is in this case the radius vector from the point where the string is attached to the center of mass of the spinning wheel and $\vec{F}$ is just the gravitational force which acts on the center of mass of the wheel. So it seems to be clear that it must point into the direction as shown in the figure.
However is it possible to see this without applying the Formula for the torque above. I am thinking of something like this:
By applying the gravitational force one would move the free end of the wheel axis down a bit for a very small time, so we get the beginning of a rotation which angular velocity vector points into the same direction as the $\Delta L$ on the picture. Thus we get an additional angular momentum $\Delta \vec{L}$ into this direction, which changes the direction of the entire angular momentum vector $\vec{L}$.
However I don't see why the last "thus" must be true, since in general the angular velocity doesn't have to point in the same direction as the angular momentum vector.
This is true for the rotation about the wheel axis since this is a principal axis of the inertia tensor (which is needed to justify that the axis of rotation must turn, when the angular momentum vector turns), but the (beginning/infinitesimal) rotation above doesn't seem to be around a principal axis.
So it would be great if someone could make the argumentation I tried more clear and more rigorous and clarify the problem described above.
Best Answer
There is a paper written by Svilen Kostov and Daniel Hammer titled 'It has to go down a little, in order to go around' that addresses precisely the question that you ask here.
The idea for the paper came from a discussion of gyroscope motion in the Feynman Lectures on Physics.
Kostov and Hammer discuss that as you say the gravitational force moves the free end of the wheel axis down a bit. They also created an experimental setup to demonstrate the dip. They found very good agreement between theory and experimental result.