Let's start by checking the weather:
Lanzarote has typical March daytime temperatures peaking at around 22C (falling back to around 15C overnight). Rainfall, at just 13mm for the month, is negligible:
http://uk.weather.com/weather/climatology/SPXX 0239
It's not much different in Gran Canaria:
http://uk.weather.com/weather/climatology/SPXX 0171
The Algarve only has temperatures peaking around 18C and falling to about 9C overnight. (The rainfall is slightly higher than either Lanzarote and Gran Canaria but still quite low):
http://uk.weather.com/weather/climatology/POXX 0013
To get really warm (or hot) sunshine in June, you're going to have to go further south than any European destination. For cheap prices that really limits you to just the Canaries. (They might be 'European' politically and culturally, but geographically they're African).
The best I can see at the moment (avoiding Tenerife) is �177pp to Fuerteventura, �185pp to Lanzarote or �189 to Gran Canaria, all with Expedia (out 07/03, back 10/03).
Chris