Disclaimer: I am not a lawyer, financial adviser nor based in the US. This is not legal/financial advice.
You may be out of luck. Starting from Foreign Earned Income Exclusion on the IRS site, and specifically Am I Eligible to Claim Either the Exclusion or the Deduction?, the key conditions that appear applicable to your situation1 are:
- You must not be employed by the US government;
- Your "tax home" must be in "a foreign country";
- You must be physically present in one or more foreign countries for 330 full days in a 12 month period.
You don't indicate whether (1) applies (my guess is you are not so employed), and it seems clear that (3) is OK. However, looking specifically at the description of what constitutes a "tax home" on Tax Home in Foreign Country, condition (2) would appear to be your biggest hurdle. An example of what would qualify for FEIE is:
For several years, you were a marketing executive with a producer of machine tools in Toledo, Ohio. In November of last year your employer transferred you to London, England, for a minimum of 18 months to set up a sales operation for Europe. Before you left, you distributed business cards showing your business and home addresses in London. You kept ownership of your home in Toledo but rented it to another family. You placed your car in storage. In November of last year, you moved your spouse, children, furniture, and family pets to a home your employer rented for you in London.
i.e. a full-scale relocation for an extended period.
Furthermore, on the difference between temporary or indefinite assignment, it says (my emphasis):
The location of your tax home often depends on whether your assignment is temporary or indefinite. If you are temporarily absent from your tax home in the United States on business, you may be able to deduct your away from home expenses (for travel, meals, and lodging) but you would not qualify for the foreign earned income exclusion. [...]
If you expect your employment away from home in a single location to last, and it does last, for 1 year or less, it is temporary unless facts and circumstances indicate otherwise. If you expect it to last for more than 1 year, it is indefinite.
Taken together, to my untrained mind I suspect you may have trouble arguing that your tax home has changed, and/or has changed permanently enough to be able to qualify for FEIE. Even if you can come up with some "facts and circumstances [that] indicate otherwise", your total time away is only going to be 12 months in total, so would appear to be classed as temporary and therefore non-qualifying.
As discussed in comments, the final clincher as to why I believe you won't qualify for FEIE is that it seems very unlikely that you will establish a "tax home" in any of the 12 countries you will be visiting, if you're only going to be in any one country for a month (but that doesn't mean you won't have a tax liability in those countries). If you never establish a "tax home" elsewhere during your 12 months' of travelling, logic dictates that it remains where it currently is – namely the US – thus excluding you from being able to claim FEIE.
1 There are some alternative ways of qualifying, but those don't appear to apply to your situation.