Some have mentioned pointless sentences, but I don't think that is such an issue for modern literature. Modern prose mostly has to do with what words come after each other on a small scale, such as a single phrase, or the associativity between words and between sentences, and at any given moment most words are only used for a few sentences, then replaced by other words.
The main problem is therefore large-scale structure, chapter to chapter. For one, the recurrence of a theme throughout several volumes of work is crucial to creating its emotional value and cohesiveness. None of which appears in Markov Chains.
And secondly, possibly the greatest shortfall of Markov Chains (ignoring performance) is in plot. They're just not very imaginative. Prose, let's say later than ancient epics, has interesting, recursive plot devices. Plot is what gives emotional power to sentences, and most markov chains are painfully incapable of creating any.
That being said, I don't think this problem is inherent to the problem of writing prose using Markov Chains. Right now it sounds like what you'd get with a crowd-sourced writing prompt, but Markov Chains can go beyond this - and literary criticism is what it will take to bring Markov Chains up to the standard of popular writers like Stephen King.
It's pretty clear that in the near future, we will laugh at the idea of reading something that someone hand-typed from their own thoughts, instead of simply reading the output of one-dimensional random walk. I hope if I have grandchildren who are as interested in modern literature as I am, their degree will be a bachelor of science, with the only work they have to do being to quantify and remove any remaining guesswork from the science of computer-generated writing. Even today, it is hard to understand why anyone still reads writing written by someone. It's as quaint as a telegram.