As far as I know, the US has a pretty unique tax system which requires Americans to pay tax even if they aren't resident in the US. (This is opposed to the more common tax systems around the world which only require you to pay tax if you are legally resident in the country - i.e. living there for six months or more per year.)
However, Bahrain does appear to have a double-taxation avoidance agreement with the US, so I'm not too sure how that would affect the tax situation as a whole. I found a document on the Bahraini government's website detailing the agreement, but it's a lot of legalese and financial jargon... Here's the link anyway if you want to take a look: www.mof.gov.bh/showdatafile.asp?rid=47&ftype=file1
We also have a page you may find useful: Banking, Money and Taxes in Bahrain. There's a short section about taxes near the bottom of the page.
Ideally you should consult an expat tax professional for detailed information on this issue, although hopefully someone in Bahrain with personal experience in this matter will reply to your question here to at least give you some idea.
Hopefully this all helps you somewhat.
When will you be moving to Bahrain? Have you had a look at potential areas where you might purchase your home?