Firstly I suggest going to www.alsa.es and put in your requirements. This website is is very straight-forward and easy to use.
It takes approx 4 hours and 45 minutes to get between Alicante and Granada and costs between 27 and 34 euros - one way (correct as of July 2009, this is subject to change).
An example of a timetable is outlined below, correct as of August 2009
If you are travelling to Granada from Alicante then please check the timetables at www.alsa.es especially because the buses travel different routes depending on the time of day that they leave.
There are about 6 buses a day to Granada and the first bus leaves Alicante destined for Granada at 02.21 and the last bus is at 21.44
If you are planning on taking the ALSA bus then please do check the timetables on their website for up to date information.
Further information on Alicante Buses
Alicante to Benidorm Bus
Alicante to Barcelona Bus
Alicante to Javea Bus
Alicante to Valencia Bus
Alicante to Granada Bus
Alicante to Torrevieja Bus
Alicante to Denia Bus
Alicante to Madrid Bus
Alicante to Almeria Bus
Alicante to Altea Bus
Alicante to Murcia Bus
Alicante to Calpe Bus