That comes with caveats as well. The service member CANNOT do so in an official capacity or in uniform (see http://en.wikipedia.org/wiki/Hatch_Act_of_1939 for more information). Political speech is even more restricted by tradition. Technically, as long as the service member is on his own time, in civvies, and not attempting to attach his/her political work to his/her office or station in the military, everything is okay.
For enlisted service members, that's what generally occurs.
For officers, there's an unwritten code (and, like most unwritten codes, fairly rigorously - if unofficially - enforced) that the officers' corp should be apolitical, to the extent that there's a pretty strict inverse correlation between the grade of officer and the likelihood that they even vote. Civillian control of the military is drummed in from the first to the last, and, for an officer, voting comes uncomfortably close to having the military take control of itself.
All of which is beside the point, of course. PFC Manning's right to free speech stopped when he publicly leaked classified information. If he had problems with the morality of what he was doing, he should have addressed them through the chain of command. If that failed, he should have addressed them through his local congresscritter. If that failed, then maybe he should have realized that he was a PFC in a war zone, and maybe not privy to all of the information required to make an intelligent disposition of the classified data of which he was a custodian.