So, you're not receiving sound when your Radias/R3 is hooked up to your computer via MIDI or USB.
Many a poster has joined to ask about this problem. Well, you made an easily excusable mistake; you believed that MIDI can transmit sound. However, this is incorrect.
MIDI is a format for electronic instruments to interact with other instruments, hence the name, Musical Instrument Digital Interface. All it is is a code that transmits information from your interaction with the keyboard, such as which note you press, usage of mod and pitch wheels, and turning of knobs.
However, MIDI does not transfer the sounds of your keyboard. (some keyboards such as the Access Virus can transfer over USB, which may have confused you. However, all Korg products at the time of writing do not have this feature) So, let's say you hit a C note on your Radias or R3. Whatever is hooked up to the keyboard via MIDI, such as a computer or another synthesizer, will know that you are hitting a C note. It will not, however, 'hear' the sound from your Radias or R3.
So you may be asking, what is the purpose of MIDI in the first place, if it can't transfer sound? Well, you can use it to control other synthesizers, and even synthesizers you can buy for your computer, called 'soft-synths' (software synthesizers). In fact, many people nowadays simply buy a MIDI controller and a lot of softsynths, so that they can have thousands of sounds all available from one box, their computer.
It is also useful if you do not like the keyboard attached to your R3/Radias; you can buy a dedicated MIDI keyboard with special options such as more knobs (which can be assigned to parameters on your R3/Radias), more octaves, weighted keys, and features like aftertouch.
Now, if MIDI is only for transferring information about key presses and knob turns, you're probably curious as to how you can record the sounds off or your Radias/R3. Well, you'd do it the same way you'd record any other instrument. The old-fashioned way to do it would be to use a microphone, then record the keyboards sounds being played through speakers. However, in the modern world, there is a better way to record.
You will need to buy 1/4' cables, and have a way to input them into your computer (a sound card). A 1/4' cable is a sound cable that you probably have seen a guitarist or bassist using:

You'll need to pick up two cables if you want to get all of the left/right effects out of your keyboard (trust me, it's definitely worth the few extra dollars.) Now, you need a way to input this into your computer. If you're going mono and just using one cable, plug into the left input of your R3/Radias. If you've decided to go with the much better option of stereo sound, plug into both the left and right inputs. (NEVER plug into the headphone input except for headphones, it has a lower sound quality and lower signal to noise ratios because of the amplifier used [thanks X-trade for this info]) Now, you'll need to plug the other ends of the cables into your computers inputs.
By default, most computers come with 1 1/8' input, usually for a microphone. You will only be able to do mono sound recording with this, unless you are lucky and either have a stereo jack or 2 1/8' inputs. In order to plug in your 1/4' cables to the 1/8' jacks, you'll need an adapter:

Now, this is not an optimal way to record your sound. It's always better to have two 1/4' inputs for your two 1/4' cables. If your computer does not have these by default, your best option is to purchase a sound card.
Sound cards are computer hardware used to record and play sounds. Your computer has one, by default, built into it by whatever company you purchased it from. However, it is generally weak, and won't have the necessary inputs you need to record your instrument with quality. You can purchase an upgrade to your sound card, though. There are two options available: External, or internal.
External are the most consumer-friendly type of sound cards. They usually use up a USB or Firewire port, and after a simple driver install you can connect your speakers and keyboard into it and voila, you're done, sound comes out of your new card. External cards also have a major advantage over internal ones; they can be moved around so as to avoid electrical noise from other electronics. However, there are two major disadvantages of an external card; they usually have slower latency (can't process sound as fast, although this has improved over the years) and are much more expensive. An external card I like is the Fast-track Pro:

Next up is internal cards. These are only for the tech savvy, or friend of someone who is tech savvy, as they will require you to open up your computer. These cards usually install inside a PCI or PCI-E slot on your motherboard, so you'll need to make sure you have a slot available. Since these cards connect directly into your computers motherboard, they have much lower latencies than externals. They also tend to be much cheaper, and the technology has been mastered for years; you can buy a card thats been around for 10 years (and is priced with that in mind) and still get a good recording out of it. Their disadvantage is that if they are near a noisy card such as a graphics card, they may pick up electrical noise, ruining your recording. However, some internal cards fix this problem by having a breakout box for your inputs, allowing you to have the latency of an internal and the noise reduction of an external. A card I like is the M-Audiophile 2496 (although you'll need to buy a 1/4' to RCA adapter):

After you've chosen a soundcard type, take a look at its specs. Make sure it has a decent mixing rate (I like 96khz and higher), that it has two sound inputs for your R3/Radias (Make sure it's two; even if it costs you a bit extra, it will definitely save you cash in the long run! Also, I like 1/4' and RCA connections best for quality), and finally, that it has the right output for your speakers! (you can't use your old sound card and new one at the same time, so this is important) Once your new card arrives, make sure to install your drivers, set it as the default card (in sound settings for an external, in the BIOS for an internal) and you're good to go!
Well, that's all I have for now. I hope this helps. Remember, if you're still having trouble, feel free to ask me or anyone else on this forum for advice! We're glad to help.