It can indeed be a risky business. My worst experience in this regard was winding up in a HAIL of bullets during javelina season in Arizona... Freaky.
Two strategies help on this score: 1) if you're hunting early in the season (like the first weekend) hike many miles into the woods and well away from roads -- more than 10 miles -- a couple of days before hand. Many hunters (in non-ATV areas) don't want to go that far in (and thus have to carry the meat that far out) and you're more likely to have an unmolested hunt. Just remember, packing the deer and your gear out will be work, so travel light; 2) if you want to hunt closer to the road don't go the first weekend -- go the middle of the second week of the season. Fewer hunters. The game will be a little scarcer, but that's OK.
The idea that the adrenalin affects the taste is generally considered untrue, although I know some old timers who absolutely swear by it and far be it from me to tell an 80 year old deer hunter what's right
In any event, chasing a deer you've shot isn't going to appreciably increase the deer's fear: it's been shot, right? Getting shot is scary enough, being chased won't make it much worse.
However, I do think -- and there is some evidence to support for this -- that the lactic acid that builds up when a deer runs can affect the taste. And because the deer dies, the lactic acid isn't flushed from the muscle by the blood. This is part of the reason you want to bleed the deer out as completely as you can when field dressing it.
At the height of the rut, bucks live a very active life and have more lactic acid in their muscles as well -- that could account for the "rut flavor" some people notice.
In my experience, deer that run a lot before dropping can taste somewhat more acrid than those that drop right away. (Remember, I don't eat the deer anymore, I give the meat to friends. A vegetarian hunter... There is something wrong with me...)
However, good field dressing makes a huge difference. The key thing (other than not getting urine or intestinal matter on the meat) is to get the animal as well bled as possible and most importantly: get that meat cooled fast.
After field dressing, I typically hang and skin the animal immediately and have found this makes a difference in flavor. A skinned deer cools many times faster than an unskinned one. Make sure to have a game bag or two handy to put the meat in to prevent flies from getting to it. A lot of hunters don't like to skin deer immediately and I think this accounts for their experience with gamey meat -- it took too long to cool.
The only time I don't skin the deer right away is if I think it's too big for me (and sometimes a buddy) to carry back to camp and thus I'll be dragging it. If I drag it (an unusual event) you want to leave the skin on, obviously. But you should still dress it and hang it to cool a while (prop the cavity open with a stick).
If you're more than an hour from camp, let it hang and cool at least 10 hours before carrying it back to camp. If you immediately truss up the deer to go back to camp it doesn't cool properly.
While it cools, I'll usually flesh the hide some so it weighs a little less when packing out (I take the hides for buckskin, always). If I'm really far in, I'll usually butcher the deer in camp after it has cooled and before packing it out so that I'm spared the weight of bones, etc.
A friend of mine insists that massaging the meat for 20 minutes or so when it's first hung makes a difference too. It might have to do with working blood out of the muscles. I tried it once and my friends said the meat tasted great. But I don't do it any more because I feel like and idiot standing there massaging a dead deer in the woods
Chinkapin, the benefit of the "sweating" you refer too is probably a cooling thing as well.
There is a cooking technique that can be effective at reducing the gamey taste: a day before cooking (or two days for a big cut), soak the venison in a bath of salt water spiked with a good amount of white vinegar. Every 5 hours or so change the bath. It will be red at first and then eventually pink/clearish. When that happens, nearly all of the game flavor will be gone.
As Chinkapin mentions, what a deer eats certainly can affect it's flavor too. But proper field dressing and cooling really make a difference.
Also, those big bucks with big racks always taste gamier than smaller bucks and does. Leave those big boys alone and alive to strengthen the gene pool and take the smaller bucks instead -- they're better eating anyway. Trophies aren't interesting, hunting is.
Lastly, Chinkapin mentioned chronic wasting disease (CWD) a few posts ago. For those of you from across the great water, it's a disease not unlike mad cow (you know a lot about that in the UK, eh?) and it's really affecting deer/elk populations in Colorado, Wyoming and Nebraska. And sadly, I've read reports that it's been found in Chinkapin's Kansas too. We've even had a scare or two in California, but happily it's not a real issue here yet.
It's very bad news and there have been fatal cases of it crossing the species barrier and affecting humans who eat infected meat. People who hunt in those areas need to be pretty careful about what animal they take....
Sorry for going on so long, but I figured some might be interested in this. Any other deer hunters (oblio13, pete79, others?), please weigh in on your experiences in this regard.