How can I receive FM radio broadcasts on my iPhone 4?

The iPhone 4 does not have a radio tuner. All it has is the appropriate cellular band, WiFi, and Bluetooth. In short, no there is no easy way to do this.

Some radio stations make apps so you can stream their content via the internet, or offer live streams. I would check out the radio station's website to see what they have available.

Lingo provides an iPhone cover which also has a FM/DAB/DAB+ receiver and extended battery pack built-in.

This looks like a great solution: Griffin iFM Radio Receiver

Not everyone may need or want FM radio capability with their iPhone or iPod, but for those that do, this is a simple, straightforward solution. Easy to install, set up and use, and according to my wife (who is the main user of this device in our household), it doesn’t seem to drain the iPhone’s battery any faster than with normal operation. The only downsides seems to be the lack of multitask (which should be correctable through a software update) and the situation with answering phone calls.

If I get an iPhone, I will get this as well.

It was released in 2009 and the last documentation I can find shows that the Griffin iFM Radio Receiver for iPhone and iPod did work with iOS 3. You should do more research or be OK with it not working as I can't be sure it's compatible (but don't see why it wouldn't be) with the iPhone 4 as a dock accessory.