One state is unicameral, Nebraska.
A parliamentary system doesn't work with the US's fixed election years, and while states don't have to use fixed election dates (as some do odd year elections) and set terms the idea is too ingrained into Americans' mindset. Plus most Americans aren't familiar with a non-directly elected head of government, even if it's not too dissimilar from the Speaker of the House.
Some countries with a parliamentary system have fixed terms, e.g. Norway, and they seem to do fine.
