Hi there. I'm in the process of looking for work in the UAE too. I trained in Australia, and now living in Canada.
I am happy to share some stuff I've found out.
Most "agencies" have no vacancies in Dubai for RN's at the moment. The hospitals there currently are hiring from India, Lebanon and the Philippines for their Nurses. This is much cheaper for them.
Abd Dhabi (about 1.5hrs drive from Dubai) has more job opportunities for western RNs. The big hospital there is Shaik Khalifa Medical Centre - think it's about 1000 bed establishment. Hospitals in the UAE: http://en.wikipedia.org/wiki/List_of..._Arab_Emirates
I would not use the "agencies" and contact the hospitals directly (or contact the company the hospital is managed by). A lot of hospitals in the UAE are managed by western companies (US, Canada, etc). For example, the Shaikh Khalifa Medical in Abu Dhabi is managed by Cleveland Clinic (http://my.clevelandclinic.org/default.aspx
Salaries for Staff Nurses are very low in the UAE. It is only decent for Senior Charge Nurses. You could expect something along the lines of 8000 UAE dirham a month. However, with your experience in your area you may qualify for a senior position (Charge Nurse, etc) - better $$$$
Most RN's with under 12 years experience that head to the Middle East go to Saudi where the salaries are much better. Around 12500 Saudi Riyal a month.
Also, Saudi offers a 1 year contract, UAE has minimum 2 years.
I'll keep you posted on more as I find out. My feeling is that if you want to work in Duabi you will want a more senior position than Staff Nurse, purely financial reasons. If you can't find a senior position in Duabi, I would suggest Abu Dhabi or Al Ain for work