Málaga
No, really, bear with me...
A week would give ample time to do a loop from Málaga, taking in Ronda, Zahara de la Sierra, Seville, Córdoba, Zuheros, Granada (book in advance if you want to go into the Alhambra) and Antequera (which is nice as the tour buses don't stop there). Jáen as well if you have time.
If you like flat, there's the Via Verde del Aceite which is a long distance path on a former rail route between Jáen and Puente Genil, kind of crossways to the circuit above, but very interesting.
Weather should be perfect in March.
Without a car is fine if you want to not-bike some parts, local buses are frequent, cheap, and have room for a couple of bikes.