Highflyer999,
You have only to go to Middle East forum and do search on this. There has been much speculation about the safety or otherwise of Qatar Airways over the years. You will need to make your own opinion on this!
Did this trip myself about 2 years ago, and had no problems at all. The changeover in Doha went smoothly, even got back on same aircraft going out, but different seat!
Not sure when you are travelling, but sometimes Qatar not the cheapest fare, especially if you don't have to travel at peak times. For instance, if you go out 13/14 March, for a week, both Air France and Luthansa have fares around 275GBP from London, whereas Qatar want 332GBP(all include taxes)
March is a busy time for Dubai, you may not find too many good fares at peak periods. Do your searches carefully before committing. At least try:
http://www.expedia.co.uk
http://www.travelocity.co.uk/
http://www.travelselect.com/cgi-bin/starter.cgi
Good luck!