Ok lets try some possibles.
Virus checker, have you excluded the FSX folders from being scanned by your virus checker? If it's loading up something (an airport addon) it might be scanning it as it loads up, as with all new scenery?
Overheating graphics card or CPU under load? - Check for dust build up in the fans and heatsink.
Ummmmm are you flying the same routes when it happens or any route?
When was the first time you noticed this and what have you installed since? Tricky one to think of I know.
Have you updated your graphics drivers, reinstalled drivers like DirectX recently, if not do.
Are you running FSX Acceleration? I am and had problems with FS Water Config, until I put the disc in and did a repair, it seemed to do a lot of repairing and my system is now fine FSWC is perfect.
Perhaps if you have the disc space Copy your entire FSX folder (everything) install to a backup folder and put the FSX disc and acceleration disc if you have it in and run repair. If it works good, delete the backup. If it messes up (which really it shouldn't) delete and move your backup across.
Other than that I can't think of anything else at the moment.
Ps. I have a very similar system to your's but have an Ati 5870 1GB HD card and I get higher fps from the start. Could you do some screenshots of your settings like Lous has above.
These are mine (I'm also using REX+OD for weather, UT2 for air traffic and FSWC for water).





All card settings to "use application settings".
The only thing that I think should be changed on mine is the "unlimited" FPS target frame rate, I think I used to have that set to lower (recommended) but might have accidentally changed it to unlimited. In fact seeing it I have just changed it to 35 and reposted the image above as I believe it frees up FSX for other things I think, plus 35 is a good rate the eye won't pick up anything faster anyway, only changes and if you set it lower than you think it can go there shouldn't be any noticeable slow ups.
And then remembering to rest the following lines in the FSX.cfg after making any in game display setting change or scenery change

[GRAPHICS]
TEXTURE_MAX_LOAD=4096 (defaults to 1024 when settings changed in game)
[TERRAIN]
LOD RADIUS=4.5000000 (default) change up to 7.50000
(gets reset to 4.5 when scenery is added or removed)