Published online in the European Society of Cardiology is a great summary document for doctors, but potentially readable for patients, of how to get started on aerobic exercise when you have heart disease.
Firstly, they summarize a very important observation which can seem paradoxical. While exercise readily and dramatically lowers the risk of cardiovascular events, the act of going out for strenuous exercise (ie: running) TEMPORARILY slightly raises the risk. This is because patients with heart disease (silent or not) have two problems. Their hearts are a setup for lethal arrhythmias (ventricular tachycardia) and rupture of a coronary plaque (causing a heart attack). The risk of both of these is increased due to more adrenaline circulating in the blood and the greater blood flow across plaques. So while the risk of having an event DURING a run is increased, the overall risk is still very low, about 1 in 200,000 patient-hours of exercise (ie: 200,000 patients exercising for an hour each, or one person exercising for 200,000 hours, talk about a long run!) For comparison, being out of shape puts you at a 4.5 fold risk of death as compared to someone who is physically fit. Not to mention, being active may reduce your risk of diabetes, high blood pressure, and metabolic syndrome by up to 25%.
They recommend a variety of tests prior to starting exercise, such as an electrocardiogram (ECG), echocardiogram (echo), and treadmill stress testing. Each of these is meant to assess the fitness of the cardiovascular system. They also recommend examination of the lungs (pulmonary testing) and joints, since many people fail to stick with exercise due to being short of breath or knee/hip/foot/other pains. The authors also provide a handy chart for putting people into low, medium, and high risk categories based on how all the tests come out. Figure 1 from their paper shows part of why aerobic exercise is so important for cardiovascular health. Stroke volume, or the amount of blood pumped with every heartbeat, goes up dramatically with dynamic exercise (ie: running) but barely moves with isometric exercise (ie: lifting weights).
In terms of what they recommend for exercise, they note that anyone who has had a heart attack or other significant heart disease should ideally first go through a supervised program of cardiac rehab before starting exercise on their own. They recommend that for people doing exercise for "primary prevention" (ie: never had a heart attack), or only have high blood pressure (hypertension), it is fine to go ahead as long as the previously mentioned testing is normal.
So, no excuses, get out there and get started walking, running, or whatever!
Also, see this page on exercise from the medical journal Circulation which is written for patients.