There is a lot of bypass surgery that goes on not just the heart. Specifically for the heart the radial artery used to be used but it is being used less and less because the success rate is lower than vein grafts. This is because the diameter of the radial artery is so small that any narrowing (for example at the ends that are sewn into the original artery, or the anastomoses) can cause failure quickly. Veins are generally of larger diameter and have a better latency rate.
Another responder to this question mentioned the internal mammary artery being used, which is true but a slightly different scenario. The area that the internal mammary artery serves is really close to the heart and so only one end (the distal or far end) is unhooked from where it is going and plumbed into the coronary artery past the coronary blockage. So the blood flows into the mammary arter as normal but is diverted to the heart.
The biggest advantage of veins is that in the body, mostly in the peripheries, eg, arms and legs, there are two sets of veins that run in parallel returning blood to the central veins (in the abdomen or chest). Think of varicose veins in the legs, these are removed surgically and the venous drainage of the leg is not harmed because there is another vein running parallel much deeper in the leg that can do the job (as it has only been doing half the job up to now). So the saphenous vein (mentioned in another reply) is in the leg near the surface and, if in good condition, can be used for a graft (bypass artery) as it is long (to get the best bit) and of large diameter so less chance of blockage. The same is true of the superficial veins in the arm (the one you see on the thumb side of your wrist is the arm equivalent to the saphenous vein in the leg).
Arteries are not generally taken as, as a rule, they do not run in parallel like the veins. The radial artery is a sort of exception because the radial artery and the ulnar artery (you can just feel it on the little finger side of your wris just a couple of inches from where you feel the radial artery) join up in the palm of you hand in a kind of arch that forms an inverted ‘U’ shape just before the fingers start. If this arch is intact (and in a minority of people it is not) you can use the radial artery because blood can flow round the arch from the ulnar side to supply the bits you have just ‘cut off the blood supply from’. It is important though to check the arch is intact before surgery, a simple but critical test.
One downside to the use of veins is that the have one way valves (which is why blood doesn’t just gather in your legs and which have failed in varicose veins). These valves either need to be disrupted (the chosen solution generally but does damage the inside of the vein which can cause clotting) to prevent the arterial flow in their new location on the heart from being stopped, or the vein needs to be turned around, so the valves are open in the direction you want the blood to flow in the heart. Veins to do have their issues though and can become really thickened and fail amongst many other possibilities.
In the long term, the grafted vein responds to the higher pressures and becomes structurally more like an artery.
TL:DR; Arteries are used but are being used less and less. Veins are used because quite often the occur in parallel pairs in the leg and arm so one can be taken and used as a graft without harm to the arm or leg. Arteries are much narrower and grafts fail more often.
Edit; Typos and clarity (yes, clarity)