You'll definitely have to work on it to make it happen - i.e. become active on those sites. However, I only have 288 karma points on reddit and I can get a reasonable success rate when I submit stuff there. The longer your account has existed and the more active it is, though, the better, of course.
Digg is a pain in the ass in that respect. The volumes of "diggs" required to even make a blip are so huge that you'll only make it there if either you've totally gamed it (like that guy who posted that article about it recently), or if you've made it through somewhere else and you're dugg as a side effect. I stopped bothering with it because I kept hearing that the quality of the traffic is also very low anyway (very high bounce rates).
Reddit and HN, however, are relatively easy to promote to, and if you get really picked up very well on reddit (which should happen with 1 article every few months if you're good and write reasonably often), it'll spread to other places too. Articles that succeed only on HN don't seem to spread around too much, however.
The best place so far, though, was Slashdot. Get yourself slashdotted (on their main page), and you can expect 100k+ hits from it, and many links from blogs, and probably you'll get reddited, delicious'ed, dugg, etc, soon after. Be sure that your server is ready to handle it... It's a complete lottery - an actual editor has to decide whether to push your article - but if you win it, it's worth it - so submit there when you have good slashdot content, it's free and might bring you untold amounts of geeky traffic!
Bear in mind also that none of these methods guarantee anything - not even traffic, and definitely no guarantee that people will stick around. But they're not overly hard to achieve if you're a decent writer and have something reasonably interesting to say (those are the hard bits, really).