In some countries, hotels and shops may accept payment directly with travellers cheques. (Once again, you sign them a second time as you hand them over). In most parts of the world, however, you need to find a bank or bureau de change to convert your cheques to cash first.
If you lose your cheques, or if they're stolen, you simply report the loss to the cheque company, quoting the serial numbers, and they'll issue replacements. (If your cheques are issued by one of the 'major players', such as American Express, they can normally arrange for you to receive replacement cheques, almost anywhere in the world, within 24 hours).
Before the invention of such things as cash machines, credit cards and debit cards, almost everyone who travelled abroad would take travellers cheques. They're far less popular nowadays because it's usually simpler and less expensive to use a debit or credit card in a cash machine. However, there are still a few destination where you can't guarantee to find easy access to a cash machine; the Dominican Republic might well be one of them.
http://www.americanexpress.com/uk/tc/index.sht ml
Chris