Agreed, the centre is good but noisy. The South side is nice - Bruntsfield, Morningside, Marchmont. I wouldn't bother staying near the airport - it is pretty far out and there are great bus links to the airport from town. Leith is OK - a bit "docklandsy" but a fair few rough areas.
Two good letting agents:
http://albaresidential.co.uk/
http://www.braemore.co.uk/info/