I'd highly recommend anyone to start from the 'bitch' position (whatever position you get owned in most) and just do work. For me it was from half, or side control. (still working on side escapes/sweeps/counters etc heh).
Missing posts moved here: Excised from AGD, the cull. - No BS MMA and Martial Arts
I'm a big guy and find that I favour sweeps from underneath far more than subs. I think that it's a combination of most big guys having relatively short limbs and the fact its harder to get your hips really moving, further evidence for that is that the exception for me is collar chokes which I can pull off fairly frequently.
It may possibly also be the fact that subs are generally easier to pull off in training, by this I mean that I see people drill submission defences a lot but it's fairly rare to see people drill sweep defences in my experience.
So I'm one of those long Ultra Heavyweights that you were talking about. 6'4" and 230lbs. I work the guard game a lot of the time in practice on purpose and I feel very comfortable there. I even prefer guard than mount most of the time.
The submission I hit the most frequently from guard is the kimura. I've submitted two guys at tournaments with that. I also have luck with the arm bar and triangle in class. Nothing compares to the kimura, though. That's my bread and butter move from any position you can hit it from.
Like a lot of other guys said, though, put yourself in the situation you dislike and force yourself to get used to it. My least favorite position is bottom of side control. I always struggle there and waste a lot of energy. I'd rather be mounted than bottom of side control. So here lately I've been working that when I roll with white belts. Hopefully I'll start getting more fluid at it.
One of the long limbed light types here, seem to get a lot more subs than sweeps from bottom.