Copper fill / Ground plane - Grounding

I am attempting to mill a PCB and to save time / tooling, I was thinking of having a copper fill on the board. Is it really necessary to connect all the fill areas up and actually ground them? The circuit is a couple of constant current LED power supplies, and also routes a HIGH (4.5v) LOW (0.2v) signal to a pin header.

It's advisable, to prevent EMC problems. For a prototype, it won't matter.

What would EMC problems entail on a simple circuit like this?

The isolated copper areas could act as antennas which are coupled to adjacent tracks, and re-radiate signals.

For hobby usage with low speed digital circuits it's hardly critical. It can be beneficial if you think of "everything that isn't a signal or +V Supply is a GND" since treating "what's left" as a single conductor can help with routing more than mess it up. I personally do it mainly because it looks nice.

I've never seen a low-speed digital circuit! The logic gates switch just as fast whatever rate they are switched.

Its important to have good decoupling, and that is more important that a continuous ground usually. If you are routing high current fast-switching signals then you must be more careful or the return currents will prefer to flow in nearby signal wires.

One approach is to explicitly route your ground network, and then pour ground - that way you know all grounds in the circuit are directly connected.