Runners as in, soldiers whose dangerous job it is to take and deliver messages and orders between the enlisted and command, something done in war for millennia. Hitler famously had this role in WWI. I’ve read that they were also used in WWII as radio communications proved unreliable in hilly and forested areas - when they worked at all.
In these days of instant communications, do armies still train and maintain dedicated runners to pass on messages? If so, how often are they used?
I really doubt it. Radio communications have improved somewhat since 1943.
The German military still has despatch riders on motorcycles (called Kradmelder).
And the Austrian military has them too. Here’s a picture on the website of the Austrian Army:
Absolutely.
As someone who’s not an expert on the technology, though, i wonder whether it’s possible for a technologically-sophisticated force (like the United States) to somehow block or jam the communications of the enemy, at least within a relatively small area?
Yes. Altough very rarely are people required to run with the messages these days, they will usually have some form of motor transport. It is usually much more convenient to send long messages/forms etc. that arent urgent physically, instead of electronically/over the air.
It is also much safer, in terms of com security - long messages are much more usefull for enemy decryption experts, and you deny the enemy data points for traffic analyses which he would get if you sent radio messages. (Altough todays modern digital radios with very strong ecryption and spread-spectrum radios has made both these problems much less of a concern). However, the messages delivered need to be secured with strong encryption if they are classified secret, as the runner/MO might well be intercepted en route if travelling along potential compromised routes.
The preferance in terms of security are usually in this order: personally - physically (runner or motor transport) - encrypted/secured telephone/data lines - radio (encryption/codes) - regular telephones/cell phones.
Of course, in practice everyone uses their cell phones for damn near everything, ruining OpSec all around 
Osama bin Ladin had only a select group of runners bring him all his information on USB drive. I believe he was finally found by tracing his most trusted runner.
Depending on the type of radio, it can be extremely easy for a non-sophisticated force to jam radio transmissions.
If you were to use a simple, off-the-shelf walkie talkie, your whole network could be jammed if someone simply holds down the TALK button. An unsophisticated radio network like this can be jammed by any military that isn’t a bunch of thugs in Nirvana t-shirts armed with AK-47s riding the Toyota trucks.
The communication systems of modern armed forces use a variety of methods to overcome jamming or eavesdropping, such as spreading the spectrum (so the radio doesn’t use one precise channel) and then hopping the message around different parts of that spectrum (to produce a type of encryption, basically).
The US Army has also been working on radios that form networks with each other, which can also defeat some jamming. Imagine Al, Bob and Chuck trying to talk to each other. Al and Chuck are on different sides of a mountain (or source of jamming), so they do not have line of sight with each other. But if Al speaks into his radio, it will know it has to relay the signal through Bob, who has line of sight to both of them, in order to get to Chuck. This technology, known as the Joint Tactical Radio Systems (JTRS), has been in development for years and has cost many billions, but it is finally starting to make it out to the field.