How do doctors test for coronary artery blockage?

Usually stress tests. Stress tests are used to see if an artery has severe blockage in it, which prevents adequate blood flow to the heart muscle to meet its energy needs. If abnormal, cardiac cath (i.e. "angiogram) is often needed to invasively image the arteries and see the extent of the problem.
Stress test. To test for obstructed coronary arteries we use various tests. An exercise stress test can indicate obstruction, if done with imaging either echo or nuclear it becomes more sensitive and specific. When we need to see the arteries we use angiography, either flouroscopic or computerized tomographic. The clinical state of the patient suggests which testing is appropriate.