I ... actually do agree about the "too many dice" thing. There are places in the rules where multiple successive rolls could easily be consolidated into single rolls, with re-rolls as a buff/pseudo-modifier being done away with entirely in favor of actual modifiers.
Then there are some units which just have tons of attacks and make you thus roll bucketloads of dice for no apparent reason other than that GW appears to think we enjoy rolling bucketloads of dice. I'm not six years old, and rolling cubes around on a table ceased to be inherently fun for me, in itself, a long time ago, thanks.
For example: Ork Boyz have 2 base attacks, and +1 for their pistol/ccw unless they have shootas, and on the charge get yet another attack. They are S3. They would be approximately as effective if they had S4 and one less attack, and the core rules could grant bonuses for charging other than +1 attack, such as striking first as in WHFB. This would cut the dice rolling for a charging unit of Ork Boyz in half, and wouldn't really change a great deal about the game (though units with high Initiative would get a bit worse, relatively; Initiative could be even more important for fleeing/sweeping advancing to compensate, then you have about no power shift at all.)
TL;DR: I'm not sure the rules are too complicated, but they could indeed do with fewer dice rolls involving smaller numbers of dice. There are also too many exceptions to core rules in recent codexes, which makes the game seem more complicated, though not in the sense that it's difficult to understand how things work.