This is a contentious topic, to say the least. From the length of the petition thread I can see many people here are firmly against electronic random number generators. So I'd like to play devil's advocate and propose that RNGs aren't such a bad thing after all (now I know how spokespeople for nuclear energy and vivisection feel).
Mechanical draws appeal to lottery players for several reasons: they are seen to be fair; people like to root for their numbers; watching the balls provides both tension and a thrill; and it just looks good when televised. Electronic draws, however, are viewed with suspicion; there is simply nothing to see. No matter how large and flashy the display, an electronic draw has all the appeal of a cash register.
But the main criticism of electronic draws is the most serious, and that is the question of randomness (whcih also means fairness). Not only is there nothing to see, but electronic RNGs aren't even random! For the first few decades of the digital computer era this was true. The numbers are pseudo-random, and after thousands or even millions of numbers they repeat themselves. Obviously such RNGs are useless for lottery games protected in law, with severe penalties for lottery operators who fall foul of them. But mathematics has finally triumphed, creating RNGs comparable to any natural or physical means of generating random numbers.
Modern RNG algorithms are triumphs of computation and the mathematician's craft. They have truly astonishing periods (how many numbers they generate before they begin to repeat themselves). The current best RNG is MT19937, developed by two Japanese mathematicians, Makoto Matsumoto and Takuji Nishimura in 1998 (revised in 2002 to remove undesirable results obtained from certain seed values).
For the technically-minded, MT19937 is a variant of the twisted generalized feedback shift-register algorithm, and has a Mersenne prime period of 2^19937 - 1 (hence the name, the MT standing for "Mersenne Twister"), or about 10^6000. Let's put this gargantuan number into perspective: at the rate of 1,000,000,000 numbers PER SECOND it would take 10^5983 years before MT19937 began to repeat. Our universe is estimated to be only about 1.2 x 10^12 years old.
Naturally such a powerful RNG is no mere academic plaything. It is used in quantum chromodynamic calculations, lattice field theory simulations, cosmological computer models, simulations of nuclear weapon explosions, and numerical tests of high-end supercomputers (MT19937 can even be adapted for cryptography). MT19937 has passed the most rigorous statistical tests for randomness yet devised, the toughest of which is the ominously-named Diehard suite of tests:
http://en.wikipedia.org/wiki/Diehard_tests
MT19937 represents the pinnacle of RNG science, but more modest algorithms would serve just as well for a lottery draw. One known as gsl_rng_rand has a period of 2^31, more than 2,000,000,000 numbers. At the rate of six numbers per week this would run for the best part of seven million years before it began to repeat. The human race might not even exist seven million years from now, let alone lotteries...
So, modern RNGs are no longer anything to be afraid of. The security issues are exactly the same as for any mechanical draw (locks, safes, and more than one pair of eyeballs). Several countries around the world use computerized draws. One of the largest is in China. Also, some games actually employ an RNG so players can get their numbers at the point-of-sale terminal. Nobody seems to mind this.
Finally, for the curious (or just plain masochistic) Matsumoto and Nishimura's original 1998 paper on MT19937 can be viewd at:
http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/ARTICLES/mt.pdf