It depends on how you arrange the lights. In the UK, there's a delay in between one set of lights going red and the next going green. In a number of US cities that I've visited, one set turns green at precisely the same instant that the other turns red. This means that going through the lights as they turn red is potentially very dangerous, because you will still be crossing the intersection while cars from other directions go. Adding a small delay, larger than the grace period, would likely improve safety considerably.
The USA has 7.1 fatalities per billion km driven, whereas the UK has only 3.6. It's tempting to blame the drivers (and the difference in driving tests in the two countries lends some support to this), but the road designers have a lot to blame. The US statistics are likely even worse for in-city driving, because the totals are skewed by the fact that you can drive far further in the US without encountering another vehicle than in the UK.