Not true, even if your doctor told you it's true. It's possible your provider will use HIPAA as an excuse, but HIPAA does not prohibit the use of email between doctors and patients.
HIPAA requires only that health information be safeguarded, and the regular email that we use everyday is not safeguarded at all.
There are programs that exist to ensure email is safeguarded. For example, some email programs will "encrypt" an email before it travels through the internet, turning it into unreadable code until someone who has the key to unlock the code receives it. Others set up systems that alert their patients that a message is waiting for them on the doctor's secure server. In both cases, all the information patients need to be able to read a secured email from their doctor is provided ahead of time.
However, for too many providers, and like with other aspects of this set of laws, email security requirements may be more than they want to handle, and they may use HIPAA as an excuse to not exchange email with you.