If cash - there is the risk to carry it. The benefits is to carry just enough to go by daily and purchase as necessary; with cash comes the benefit of purchasing power everywhere.
If ATM Card - there is the risk to lose it, have it eating by the ATM machine or cloned. Benefit is to be able to draw cash as needed.
If US CC - the risk to have it lost, stolen or cloned. The benefit is to have the CC backed by a US institution with customer service to respond to any fraudulent charges, if any. Also, if you're into rewards from the US CC and planing to spend lots of money, you can gain rewards points on the purchase.
If Local CC - the risk; must go through the approval and paperwork process (tedious in the DR banking arena) and the benefits is to be able to use the local CC and deal straight with the bank should anything go wrong with any transactions.
Everything would come down to your risk/tolerance factor. My personal choice is a combination of all depending on each situation. I've used my US CC and ATM Card w/o any problems at all in the DR. Personally, me, myself and I hated all the red tape, bureaucracy and the many different standards required by each DR bank, so, no local CC for me for that alone.