The great debate: is the universe discrete or continuous? So controversial is this topic that Scientific American hosted an essay contest on the subject (Link.)
Personally, I've never been a fan of Digital Theory. While cool and fun to think about, it just seems too neat, too idealistic. An analog universe makes much more sense to me. Not only is it fluid, and thus messy, but it is also better supported. After all, as pointed out by Jarmo Makela, all we know of our discrete particles has been determine by collecting data in the forms of electromagnetic fields (something fluid).
Actually, a lot becomes significantly more complicated in the discrete case. In the continuous case you have the existence of limits, and all the fun stuff that follows from that like calculus. This puts a big constraint on the niceness of possible behavior of stuff sitting in spacetime. Whereas in the discrete case we get more complicated stuff. Notions of measurement become kind of clunky and intractable. Directions no longer make sense, or at least make significantly less sense, on the small scale. Things that would break apart into simpler pieces in the continuous case become irreducible, or only decompose in strange ways.
This is part of the reason quantum mechanics was so weird to start with, and where it got its name: things that were thought to be continuous, like energy levels, were found to be discrete, quantized. If the physics that leads to energy being quantized is so weird, we should expect physics that leads to spacetime being quantized to be even more so.
This is also the reason that statistical mechanics is often done in the large n limit, pretending that you have an infinite number of particles instead of only a finite number; finite, discrete things are harder to deal with than infinite, continuous things.
On the other hand, some avenues of quantum gravity research, specifically topological quantum field theory, seems to not care whether spacetime is continuous or discrete due to the possibility of triangulation.