Links for Airlines:
The earlier you book (a flight), the cheaper flight seat you find. It is recommended not to wait to book a last moment flight.
1) Ryanair: From London Stansted, Liverpool, East Midlands, Bournemouth, Athens, Frankfurt-Hahn, Brussels (Charleroi), Rome Ciampino, Milan Bergamo, Bologna, Pisa, Kaunas, Oslo Rygge and Stockholm Skavsta to Rhodes airport. 
Ryanair link

2) Jet2.com: From Manchester, Glasgow, Newcastle, East Midlands and Leeds Bradford to Rhodes airport.
Jet2 link

3) Easyjet: From Liverpool, London Gatwick, Paris Orly, Berlin Schoenefeld and Milan Malpensa to Rhodes airport.
Easyjet link

Tui Flights: From Birmingham, Bournemouth, Bristol, Cardiff, Doncaster Sheffield, East Midlands, Glasgow, Leeds Bradford, London Gatwick, London Luton, London Stansted, Manchester and Newcastle to Rhodes airport.
Tui UK 

5) Norwegian.no: From Copenhagen, Oslo-Gardermoen, Lulea, Stockholm-Arlanda, Aalborg, Alesund, Alta, Bardufoss, Bergen, Haugesund, Karup, Kiruna, Kristiansand, Molde, Stavanger, Tromso, Trondheim and Umea to Rhodos airport.
Norwegian link

6) Finnair: From Scandinavian countries to Rhodes airport
Finnair link

7) Transavia.com: From airports in Netherlands to Rhodes airport
Transavia link

8) Alitalia: Airline from Italy to Rhodes airport
Alitalia link

9) Tuifly.com: From Germany and Switzerland airports to Rhodos airport
Tuifly link

10) Condor: From Germany airports to Rhodos airport
Condor link

11) Eurowings: From Germany airports to Rhodos airport
12) Smartwings: From Prague, Brno, Ostrava, Bratislava and Budapest to Rhodos airport
13) Edelweiss Air: From Geneva airport and from Zurich airport to Rhodes airport
Edelweiss link
14) Jetairfly: From Brussels airport, Charleroi airport, Liege airport and Ostend-Bruges airport to Rhodes airport.
Jetairfly link
15) Aegeanairlines: From various airports in Europe to Rhodes airport
Aegeanairlines link