I have friends with a 4 bed house in Ayia Thekla who take on long term renters.

Here is the link http://www.cyprusvillas4u.co.uk

My friends actually live next door in thier 4 bed place. We visited them in May and can fully recommend it.

You could try putting a post on the Cyprus Eastern site in the property section.
http://www.cypruseastern.com

This is a site where most of the members have bougth or are buying property in the area.

There you will find a list in the services section one person worth contacting is George at Island Homes office is on the road from the roundabout going up to the Orphanides supermarket. He always has lots of proerties for long term rent.

My partner's parents are currently renting long term in Paralimni from a rental agency. We helped them sort it all, very professional and very reasonable rates with a variety of properties available in Protaras, Pernera, Paralimni, Ayia Napa, Ayia Thekla, Kapparis etc. His parents are in a fully furnished brand new 2 bed apartment in a lovely complex with communal pool for £300cyp per month.
http://www.villas2let.com.cy