Hi Lin
I'm English and have spent a total of 20 weeks in the DR on various holidays. I have always relied on ATMs (cashpoints) without problem, but be aware that scams and technical malfunctions are much commoner in the DR than in the UK.
Your card may be "eaten". For this reason try to use an ATM outside a bank in your own area and during opening hours so you can seek immediate assistance. You may have to return later to reclaim your card and will need your passport. If you use a stand-alone hole-in-the-wall ATM getting your card back could be a real nightmare.
Many of the ATMs at smaller banks have a low maximum payout (RD$3000 or RD$5000, which will not last you long). While it is sensible not to carry too much cash around, this may be an uneconomical way of drawing money if you are charged a flat withdrawal fee. I believe Banco Popular always pays out at least RD$10000 but perhaps more knowledgeable posters can expand on this?
I prefer to use ATMs where there is an armed guard nearby. Less chance of the machine being tampered with or of being jumped while I am hiding away the cash.
When entering amount, enter an "odd" amount such as RD$9900 (rather than RD$10000). This will provide you with some smaller denomination banknotes. Small change is hard to come by in the DR.
Use normal common-sense precautions like checking the machine for any suspicious looking "add-ons". Never accept help from others in using the machine. If you see persons loitering near the ATM, come back later or use another one.
Remember to carry your card's appropriate "help" and "fraud prevention" telephone numbers with you so that you can immediately ring to the UK if your card is lost or stolen.
If you have online banking facilities, check your account from time to time at an internet caf? for reassurance. ATMs often run out of cash but sometimes still debit your account, so if one fails to pay out, check your account online (remembering that it may typically take 3 to 5 days to show a withdrawal).