Almost everytime I have my mobile in my shirt pocket, someone is quick to point out that I shouldn't keep it near my heart.
Is there any link between a mobile phone and heart troubles?
Is it true?
The answer is Maybe, if you have a pacemaker, and then only potentially.
Normal people: No as I couldn't find any warnings where I would expect to find them.
As I'm not a medical doctor, if you are really worried you should ask a cardiologist. I can not be sued for malpractice (and disclaim liability for my answer), but as professionals they can be held responsible for their advice.
Now for the evidence:
Here are some guidelines for pacemaker users from a medical practice.
Cellular phones, CB radios and ham radios can sometimes interfere with the function of some pacemakers. Cellular phone and CB radio antennas must remain at least six inches away from the pacemaker. Hence do not carry a cellular phone in your chest pocket, even when it is turned off. When using a cellular phone, hold it to the ear farthest from the pacemaker generator. Ham radio antenna should remain at least six feet away from the pacemaker.
Power levels from amateur radio equipment (up to 1500 watts) are much higher than cellular phones (maybe a watt or so). However, the antenna for the 1500W radio will be at least 10 or more meters away but the cellphone can be put in your pocket, much closer to the heart , and the effect of EMF is a 1/r^2 effect. Still amateur radio is a good place to look as it has been around a lot longer.
http://www.arrl.org/files/file/Technology/tis/info/pdf/jul94.pdf notes that historical pacemakers has trouble with various EMF producing equipment such as microwave ovens, certain medical scanning equipment that produced 13 and 27Mhz signals, and welding equipment, but has not cited any information about deaths. It claims that modern pacemaker operation should be safe in most amateur radio environments that meet RF safety standards.
http://www.sss-mag.com/rfsafety_bkg.html has a nice article on RF safety. It refers to an American Physical Society publication that could not link RF fields to cancer -- but you are asking about the heart.... The article also has a section on pacemaker safety.
The only websites where I could find a connection between a normal heart and interference from radio transmitters (such as phones) were holistic in nature. There was also this spam embedded in an Amazon discussion group: Chronic EMF Increases Stress, Appetite, Obesity, Heart Disease, Diabetes, etc.
Google "American Heart Association cellphone" only brings up concern in the context of pacemakers:
From http://www.americanheart.org/presenter.jhtml?identifier=4676 Cellphones
Cellphones available in the United States (less than 3 watts) don't seem to damage pulse generators or affect how the pacemaker works. Technology is rapidly changing as the Federal Communications Commission (FCC) is making new frequencies available. Newer cellphones using these new frequencies might make pacemakers less reliable. A group of cellphone companies is studying that possibility.
If there were danger, I would expect the American Heart Association or one of the other sources to announce it.
So my final answer is: I could find no evidence of warnings about the dangers of cellphones to a healthy heart directed at the general public at the most likely authoritative source
If you are asking for personal safety reasons, you should consider that: (1) the effect of a radio transmitter is proportional to 1/r^2, twice as far away is 1/4 the signal. (2) in your automobile, an exploding airbag hitting your chest pocket may break the phone up into sharp pieces that could hurt you.... (3) there have been reports and demonstrations (e.g. on youTube) that under some circumstances Li-Ion batteries can ignite... and most phones use Li-Ion batteries. However, there could be some warnings... like smoke and heat... and isn't common. If any of this really worries you, just put it in another pocket