Of course, while this is a very humane way to execute an euthanasia (either of animal or a human), talking about the least inhumane way to carry out an execution is somewhat ridiculous. The most humane way is to just not do it.
Also, it's totally off-topic.
As far as I can tell from my hasty research, all states in the USA claim lethal injection as their primary method (note: 17 states do not practice capital punishment). A few still have electrocution, firing squad, hanging, and gas chamber on the books, but they seem to be vestigial laws or "backups" in the event that lethal injection is deemed unconstitutional.
That said, according to this site [0] there have been recent executions by firing squad (2010), electrocution (2010), hanging (1999), and gas chamber (1999).
[0] http://www.deathpenaltyinfo.org/views-executions?order=exec_...
The morality of the death penalty itself is a reasonable thing to argue over. The idea that some current implementations (chair, gas) are better than firing squad (or hanging/beheading for that matter) seems nuts.
Of course, given the choice, I'd probably choose to go the same way as my dog went. Sleep.