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 1 hour and 10 minutes to get between Alicante and Murcia and costs a very resonable 5.38 euros - one way (correct as of July 2009, this is subject to change).
There is 1 bus an hour to Murcia and these leave on the hour.
The first bus leaves Alicante destined for Murcia at 7.00 and the last bus is at 21.00
If you are planning on taking the ALSA bus then please do check the timetables on their website for up to date information.
NOVELDENSE, LA bus also travels to Murcia but you would have to call them directly for 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
Alicante to Calpe Bus