Does H3O+ donate a H+ to Water?
$\ce{H3O^+}$ is really just a "shorthand notation" for what really occurs in solution.  Typically higher hydrates of the proton, particularly $\ce{H5O2+}$ the "Zundel" cation and $\ce{H9O4+}$ the "Eigen" cation are thought to be the main players in the proton transfer process.  As permeakra has pointed out, Grotthuss was one of the early pioneers studying this process and his name is still used to describe the mechanism.  The mechanism is still an active area of research with considerable controversy remaining over the details of the mechanism.  This reference provides some diagrams of the general process.  



Basically, just as electrons are said to use a "hopping" transport mechanism, the same has been said of proton transport.  The making and breaking of hydrogen bonds, accompanied by the twisting and turning of protons completes the process.  This proton hopping mechanism between these larger water aggregates is thought to be a key step at room temperature.  However proton tunneling plays a role as well, particularly as the temperature is decreased.