The answer depends on latitude, so let's look at the worst case along the equator. At the equator, the rotation of the earth gives us a linear motion of 1670 kilometers/hour, or 463 m/s. If you reduce this to 0m/s over the course of an hour, that corresponds to a deceleration of 0.129m/s^2.
This is a small number by human terms. A car decelerates at shy of 10m/s^2, nearly 100 times more aggressive than our slowdown will be. This rate is roughly 1/10th the acceleration you feel when an elevator starts to go up.
Things which we tend to think of as responding to accelerations will not be hurt by this kind of change. However, things we tend to think of as either solid or unrelenting may require further investigation.
This means trees wont mind anything, and neither will the buildings. Mountains would generally not be bothered (though some particularly precarious monuments may topple).
The two places I would expect to see noticeable effects is the ground and the ocean. The ground would be put under intense strain by this kind of acceleration, simply because it is a massive mass which has not settled in a direction that deals with that acceleration well. I would expect earthquakes in all of the regions traditionally associated with earthquakes, but I would not expect them to extend into more stable areas (the ground would probably be stable enough to take the beating).
The oceans are interesting because of momentum. To draw an analogy, consider one of these toys http://www.amazon.com/ALPI-Liquid-Wave-Paperweight/dp/B004P93Y78 . The effect on the ocean would be remarkably similar to tilting the paperweight a few degrees. The water rushes towards its new "level." The initial effect would be nothing more than some minimally rising tides. However, in the middle of the ocean, there would be an opportunity for the water to build up an enormous amount of momentum, just like the waves in the toy can build up. This happens because the flowing water is remarkably good at conserving energy as it shifts, so it simply continues to pick up velocity until the land gets in its way.
The worst case scenario? A volume of water hits the land just after the rotation has stopped, without losing any of its initial velocity from spinning with the planet. This would be a 463m/s tidal wave of epic proportions, causing massive damage in its wake. In reality, it won't be perfect. Interaction with the ocean floor would cause some slowing. A tidal wave tend to hit around 200m/s, so if 50% of the energy gets conserved, a tidal wave the size of the ocean will hit our seaboards, all at once.