"Playing" is the word I would use (British English).
Broadcasting does sound weird because the radio itself is receiving a broadcast, not broadcasting.
You can also say that a radio is tuned to Radio 4.
(Radio 4 or BBC Radio 4, or just 4, but not BBC 4 because that's a TV channel.)
ell.stackexchange.com/users/12600/ssav
"Tuned in" is probably the safest option, as LanguidSquid and ssav have mentioned.
"Playing" is applicable, because the word "play" in this context refers to "playback," which has a different meaning than the normal usage of the component word "play."
However, usually "playing" is used when referring to songs and music specifically, because "playback" occurs when someone starts a record track, or more commonly nowadays, a digital audio file from the beginning.
Usually talk shows or speeches are not pre-recorded, so the words "playback" and its shortened form "play" aren't quite as applicable.
However, most people don't regard such nuanced differences in semantics.
In usage, "play" would refer to the song or other audio presentation that the channel is broadcasting, not the channel itself.
"Receiving" is correct, although not used very often.
It is the most specific/appropriate word to use to describe what exactly the radio is doing, but most people leave it out of colloquial conversation.
However, you could definitely call your radio equipment a "receiver."
The transmitter or broadcasting device is somewhere else -- usually where a radio signal tower is located, in the channel's studio.
ell.stackexchange.com/users/10452/Crazy Eyes
The radio is tuned in to BBC 4.
It also makes sense to say 'The radio is playing BBC 4'.
The radio is a similar device to a record or CD player and these 'play music'.
Similarly, a television that is tuned to/turned to BBC can be described as showing BBC when the channel is set to BBC.
ell.stackexchange.com/users/24552/Blake Yarbrough
I would make a distinction.
Tuned to X or just set to X means the knob(s), switches, button(s) or whatever are set to a particular frequency or station X rather than some other frequency or station.
But the radio might not actually be reproducing the sounds broadcast (or transmitted) by X because: the radio is turned off, or does not have a working source/supply of electric power
X is not transmitting right now,
for example shut down overnight the radio is far away from X, farther than the radio signal can propagate
the speaker(s) and/or headset is(are)
disconnected or broken Receiving X or playing X means the radio is tuned to X and "on" and X is broadcasting and the radio is reproducing the sounds being broadcast by X.
ell.stackexchange.com/users/19136/dave_thompson_085
I know you're asking for a verb, but since a couple people have suggested "tuned in", which is not a verb, I would like to suggest that a verb is not needed at all, and there is a very common way of describing what is on the radio (I just used the phrase myself without even realizing it).
In American English, you can say, "The BBC was on the radio" (meaning the BBC was playing on the radio) and also
"The radio was on the BBC" (meaning that the radio was tuned into a station broadcasting the BBC).
Here is an example on the web: "The radio was on BBC stations all day..."
And here : "The BBC was on the radio..."
Talking about something "playing" on the radio sounds to me like music was on the radio.
I think that comes from the fact that we say that a musician or a band "plays" a song or artists' work.
When a song is heard on the radio, then the radio is "playing" it.
So, you wouldn't say that the radio is playing the BBC, because that's likely not a musical program, but you would say that the radio is playing Beethoven's Fifth, or Led Zeppelin, for example.
However, music and songs can also be 'on' the radio.
ell.stackexchange.com/users/9675/user151841
As others have said, there are quite a few options depending on the context.
Here are some specific examples that show how different words apply: I couldn't hear you before because Ron had the radio going .
So NMR played this story on the radio earlier about a man and his store.
I can't get the radio to pick up a signal in the shop.
Stupid teenagers think driving around with the radio blasting is cool. ...
and there's a lot more
and they're all a little bit different.
Going implies the radio was background noise to the speaker (he wasn't paying attention); played implies whatever was on the radio was prerecorded; pick up specifically refers to the radio getting a signal; blasting refers to the volume of the speakers, even though we apply the verb to the radio.
TL;DR radios "do" a lot of different things in different contexts - you have to find the right word for the situation.
ell.stackexchange.com/users/20104/user1717828
If the radio is on, but nobody is listening (there are people there, but they are ignoring the sound), you can say the radio is blaring .
This can be transitive ("it was blaring BBC 4") or intransitive ("it was blaring in the background").
The definition doesn't require that nobody is listening, but that is the connotation if you don't say otherwise.
ell.stackexchange.com/users/12082/Kevin
Playing.
As in "The radio's playing some forgotten song, Brenda Lee's comin' on strong" (Golden Earring, Radar Love) @Kevin: Blaring just means it's load and/or distorted.
Nothing to do with nobody listening.
http://www.merriam-webster.com/dictionary/blare .
Oh, and "in the background" doesn't make it intransitive, because that's not a direct object.
ell.stackexchange.com/users/24603/BlokeDownThePub
I'm trying to find the right verb for when the radio is ... broadcasting? ... a particular channel. In my native language, a direct translation would be "the radio was playing BBC 4" but that doesn't sound right in English. "Broadcasting" sounds weird when it's just a radio receiving signals. Receiving?  To broadcast? To receive? Something else?
Verb for what the radio does
ell.stackexchange.com
<word-request>
ell.stackexchange.com/questions/68827
ell.stackexchange.com/users/24550/Niklas
"The radio was playing BBC 4" makes sense. You could also say "The radio is tuned in to BBC4", or "BBC4 was on the radio". Playing, blaring, tuned in, on are options.
Blaring, tuned in, on are options. You can use playing.