Dubai taxis will drive to Fujairah on the meter (at least the airport ones do - its something over 250 Dhs). I'm sure they would do the same to Dibba.
Otherwise bus to Fujairah (or get off at Masafi) from Union Square, and taxi the rest of the way.
If in a hotel, I like the idea of asking hotel to arrange transfer.
Does anyone know if taxi's in Dubai have a set price if you needed to travel to Dibba or does anyone know of a company that would organise a car to drive 3 people from Dubai airport to Dibba?