Traveller's cheques are still an option but not the best option.
I finally used up my 12+ year old cheques last year. Atlantic Bank and Scotiabank would not cash them unless I had an account at the bank, and then the process was going to take about three weeks for the money to be deposited into the account -- not possible to receive cash. I cashed one cheque just for interest, and the money appeared in the account a few weeks later. No fee was charged.
At Belize Bank, the cheques were accepted immediately, and I obtained the funds in BZ$. No fee was charged. Passport required of course. Very painless process as long as the queue is not too long.
I've never tried to use cheques at a business.