The idea that Martial Damage Dice are all d6s is gone. Instead (as mentioned above), when you are meant to do more damage, you roll your weapon die more times. 1[W], 2[W], 3[W] etc. And it seems this is the case for all classes that use weapons. You get more weapon dice as you level, instead of your one weapon die plus a whole heap of d6s. Plus... these dice are only for damage-- they are not used to trigger any other abilities.
In terms of the "Basic" game... I imagine it can be treated exactly like we thought they would be previously-- instead of a random number, it'll just be a set average amount in place of each additional die. Forex: At 4th level(?), rather than rolling longsword damage as 2d8... you'll roll 1d8+4. At 7th level, instead of 3d8 you roll 1d8+8. Etc. etc.
Now for fighters, on top of those extra weapon dice... you'll get expertise dice too that you'll spend to activate all your special maneuvers. Removing damage via Parry, Raising AC, Glancing Blow sounded like it got changed to where you add your ED to your attack roll to change a miss to a hit (rather than previous iterations where it granted a small amount of damage even on a Miss). Tripping, reducing Cover, etc. etc. all gained via ED. And it did not sound like these ED could ever be spent on extra damage if they weren't used to trigger any maneuvers.