Temporary Driving Licenses

Visitors of countries which can do a straight Dubai driver's licence exchange, are allowed to drive private vehicles without obtaining a temporary licence, according to the Dubai Police website (as of April 2009 or earlier, confirmation not found on the RTA website) as long as the entry visa is issued in Dubai, and the visitor holds a valid licence.
If planning to drive in another emirate, then the visitor needs to obtain a temporary driving licence (costs 100 dhs, visit an RTA office). Before driving, confirm with your insurance company that the driver is insured. Update: the permission for visitors to drive a private car in the UAE without a temporary licence might only apply to first degree relatives of the car owner (father, mother, brother, sister, possibly children). Confirm with the Dubai RTA since rules sometimes change without notice, or are applied inconsistently.

Unconfirmed reports surfaced in December 2006 that the issuance of Temporary Driving Licenses was suspended, but this appears to have been due to confusion arising over changes in rules, or interpretation of rules. The relevant authority is the RTA (Dubai Road & Transport Authority) instead of the Dubai Police now and they appear to be implementing rules more strictly than the Dubai Police, and/or adding new rules. Confusion arises because of the different interpretations of rules and combinations of UAE visa, original nationality, original country of residence, country from which original driving license was issued.
Note that this means there is quite possibly variation in the way rules are implemented in different Emirates as the RTA is a Dubai organisation, not a federal UAE organisation.
A temporary driving license is issued if you want to drive a private car in Dubai but don't have a residency visa to be able to obtain a full UAE driving license eg visitors on visit visas, expats waiting for residence visa processing. You obtain it from a traffic police station (there's one next to Mall of the Emirates in Al Barsha) and it's valid for six months. Bring:

• Your home country passport, driving license and copies - your licence and passport are supposed to be from the same country.
• Copy of whatever visa you have for the UAE (or passport if it's stamped, or e-gate card if not) - if you have a residence visa you are not supposed to be able to obtain a temporary driving licence.
• International Driving Permit from the same country as your licence. As of early 2007 this appears to be more necessary than previously but you never know if/when rules or interpretations of them change.

Note that if you want to drive a rental car in Dubai, you don't need to apply for a temporary licence but you do need to bring an IDP - International Driving Permit (required as of January 2007). The Dubai Police website seems to say (as of April 2009) that if you hold an IDP then your home country driving licence is not necessary but bring it just in case. Rental car companies may or may not ask for both.