Does the heart only receive blood during diastole?

Short answer: pretty much, yes.

In case you're wondering why this is surprising, recall that almost all tissues in the body receive the bulk of their blood supply when the heart is contracting (that's why the heart contracts). But not the heart, which is a bit ironic.

Why? During the left ventricle's systole (when it contracts), the heart contracts with such a force that its private blood supply (the coronary vessels) is obstructed. Some blood still gets though, though, but not much.

Then, in diastole (when it relaxes), the bulk blood of the blood flow to the left ventricle rushes through. The same applies to the right ventricle, but since it is contracting with much less force, the systolic flow is impeded to a lesser degree.

The fun really starts when the diastole is shortened, as in severe tachycardia and some conditions of the heart valves. On these occassions, the heart can struggle immensely to cope with the load, resulting in anything from angina to even, if chronic, heart failure.