Over the years in different parts of the world, the Car Rental company I have used at over 90% of the time which had the best rates over others, has been with Alamo, booked directly with them online, with the Insurance included.
https://www.alamo.com/ . I used Alamo also in Larnaca Airport few years ago and they were on the premises at the old terminal. The good thing about booking online with Alamo is, although you get a confirmed booking at the price you booked it, you do not need to confirm it with a Credit card, which means, if you find a better deal later, you can always cancel the Alamo booking without any cost or penalty. You don't even need to cancel, as it would just expire if you do not show, but it is always a good policy to cancel online if you are not going to use the car. In fact, Alamo will remind you via email few days before your pick-up date if you are still going to need the car or not. Most online booking companies might show a cheaper rates than Alamo's, but often it does not include the daily Insurance on the car, so be aware, because you will get a nasty surprise at the time of the pick-up.
http://www.ebookers.com is actually one of the good online booking companies that does include the insurance with the price on the cars they offer from all the leading car rental companies, which I have also used the remaining 10% of the time, but they do require that the booking is confirmed with a credit card and that there is a penalty if you should cancel at later stage. I really think that Alamo is the tops when it comes to prices and booking rules.
Note: Just to clarify something regarding the daily insurance being included with Alamo in their prices.
If you put the United States as your residing country, often the daily insurance is not included in the rates, but the rate is also much lower, that's because, most people with Gold and higher rated Credit Cards, when are used booking with it, one can refuse the insurance on the rental car and that the Credit Card company covers you in case of a claim, as long as you sign away the refusal of CDW with the Rental Car company. But if you put any other place as your residence in the world, often you will find that the daily insurance is included in the price.