The rule of thirds is only one of a number of compositional approaches to landscape. It's only really useful for static shots with a feeling of calm and repose. And only if you have one focus.
People tend to mention it a lot in Photography because it is the only compositional 'rule' they know - and if you divide an image up using it (or the Golden Mean) then most things end up near enough to the lines to 'prove' it.
Good landscape comes from the photographer's reaction to it. Putting what you feel towards what you see - or how it makes you react, if you like - on film. Or making the viewer aware of some little detail or aspect you see and like.
It could be the curve of a hedge, the relationship of a tree to something else, or just a feeling of space.
There are many things we can react to or focus in on.
The important thing is that you feel something towards what you photograph.
If you look at a view and think 'that's pretty' then try and encapsulate that in the image.
Or if it's a wide open space that makes you feel lonely, then get that over.
Don't take pictures of views just because you think you should (unless you are doing it just to record it).
If a landscape does nothing for you then the odds are that a photo of it will do nothing for the viewer.
Remember too that there are many kinds of landscapes and they don't have to be big sweeping views. Small sections of a larger vista can work too.
And don't stand where everyone else does. Moving just a few yards can make all the difference. Even lying down or climbing a tree to change your PoV can do the trick.
Time of day can make a difference as well. Something that looks boring or uninteresting at one time of the day (or in sunlight/fog/overcast/rain etc) may be magical under other conditions. Seasons, too.
And finally, don't be afraid to keep going back to the same locations - or do reconnaissance. You might miss the magic the first time around.
