Turkish Airlines do, but you'll need to route via Istanbul. They fly from LHR to Istanbul. Their timetable is available at:
http://www3.thy.com/troyaonline/timetable.tk?lang=en
Their flights to istanbul are:
1 2 3 4 5 6 7 06:55 12:40 TK1982 738 Non-Stop 3:45 hrs.
1 2 3 4 5 6 7 10:45 16:30 TK1980 313 Non-Stop 3:45 hrs.
1 2 3 4 5 6 7 16:30 22:15 TK1992 738 Non-Stop 3:45 hrs.
Their flights from istanbul are:
1 2 3 4 5 6 7 07:30 09:30 TK1979 313 Non-Stop 4:00 hrs.
1 2 3 4 5 6 7 12:10 14:00 TK1991 313 Non-Stop 3:50 hrs.
1 2 3 4 5 6 7 19:00 21:05 TK1981 738 Non-Stop 4:05 hrs.
BA also fly to istanbul if you wanted to book the two legs seperately:
1 2 3 4 5 6 7 09:55 15:40 BA676 767 Non-Stop 3:45 hrs.
1 2 3 4 5 6 7 16:05 21:45 BA680 320 Non-Stop 3:40 hrs.
BA's return flights are:
1 2 3 4 5 6 7 08:55 11:05 BA675 320 Non-Stop 4:10 hrs.
1 2 3 4 5 6 7 17:00 19:10 1 BA677 757 Non-Stop 4:10 hrs.
Connecting flights:
Istanbul - Dalaman
1 2 3 4 - 6 - 08:00 09:10 TK 214
- - - - 5 - 7 19:20 20:30 TK 218
Dalaman - Istanbul
1 2 3 4 - 6 - 10:00 11:10 TK 215
- - - - 5 - 7 21:20 22:30 TK 219
Hope thats of some use!