No. The water itself in the middle of the ocean never moves, it actually goes through a series of small vortices and comes back to the same place it started at. The entire vortice might be only a few centimeters in diameter.
The thing is that the wavelength, the actual distance between first and second crest, and wave height are unnoticeable at open sea. As that energy wave nears the shore, it compacts to a degree and the ocean floor begins to "push" up the water as the energy moves forward. That's the critical issue, the water only moves up and inland when the energy wave begins to interact with the ocean floor.
Here's a pretty good animation showing what is referred to as "stacking" when the waves come onshore.
http://www.pep.bc.ca/tsunamis/causes_2.htm
On the bottom left of the map, click the next button 3 times until the wave starts to travel from left to right on the screen. It'll show the waves being not noticeable at all offshore, although divers in the water itself will fell the energy wave pass through their bodies. The next animation after that shows the stacking effect that causes the majority of shoreline and inshore damage.
The secondary issue of flooding can be seen by then hitting next again once that animation completes, but it's really just showing the breadth one of these waves can have and the fact that there are usually several of them.
Of note, was tsunami waves come close to shore, the sea will retreat as the energy waves stacks offshore, causing water to run back towards it. Some of the Indonesian tsunami footage showed fairly extensive beaches with no water on them for hundreds of meters offshore.