Although this is a pretty old topic, I just chanced upon it during a Google search for some info.

Here's a list of stations that are currently running in Doha:

FM Stations

90.8 – QBS Arabic
92.0 – MBC
92.6 – Radio Sawa Gulf
93.4 – Radio Monte Carlo
93.7 – QF Radio
94.0 – Orynx FM French
94.6 – Bahrain FM
97.5 – QBS English/French
100.8 – Sout Al Khaleej
101.7 – Al Jazeera English
103.4 – Quran Kareem Radio
104.0 – Radio Emirate FM
107.4 – BBC Arabic Service

 

It's not much compared to what other countries may have, but I guess the other alternative is the Podcast directory on iTunes and other such websites for a more global coverage of news, music etc.

I grew up in Doha in the '90's and the above list of stations is a tremendous improvement from the 1 or 2 stations that were available in the '90's. It's impressive that there even is a 'list' of local stations at all and that too, in different languages.