Is anything from the original trilogy going to matter?

If they did put anything in from the original trilogy you know people will just complain about that saying they want a "Fresh Start" and not be constantly reminded of past adventures.

It's already been confirmed by Mac Walters, that there will be several references to 'pre-launch' MW lore/characters/universe bits: with Shepard confirmed to be mentioned as well.  Obviously, no character decisions can be taken into account to confirm a canon lore-set, but I for one am glad they are not rebooting, but going on a wide tangent instead (by leaving the MW).

Personally, I'm glad it's a fresh start for sure, but also glad that the base lore isn't being tampered with or disrespected/abandoned in terms of any unnecessary redacting of previous world building (ME Universe Codex etc).

If anything, the reason we might not see Quarians or Geth is because of the ME3 choices. We have the option of destroying both and neither joined the Initiative. It was just the Council's people and Krogan. Other than that, anything Shep did will most likely not affect MEA.

That was a Joke......

I really doubt that what happened in the original trilogy will matter in the first game, but it would be neat to travel to the milky way at some point and see what were the repercussions of every ending, that would be a really big narrative hook there.

Post-ME1? No, it's unlikely they bring anything in unless it's inconsequential. Bioware is setting Andromeda when and where it is specifically to avoid having to deal with the messy continuity ME2/ME3 would bring along with it.

people will complain about anything.

I still think the main reason for leaving is ultimately because of the potential threat of the reapers.  Leaving the milky way galaxy before its purged makes for a good reason not to have the game set in the MY.  We know from one the first trailers in 2015 that Shepard is addressing the Andromeda crew so its possible that Shepard is one of the big players behind the initiative (Video below).  We also know the initiative is funded by big players so I think the 1st trilogy may come back full circle in later games.  The Andromeda Initiative is supposed to somehow create a bridge between the two galaxies.  My bet is we'll revisit the Milky Way eventually (again another game) or see how it has shaped 600+ years later.

Pretty simple to pick one the four endings as well from ME3 to see how it shapes out 600 years later.  And even cooler imo to eventually work side by side with the reapers (now immortal shepard in god form) to eventually eliminate some new threat hundreds of years later. (Andromeda Galaxy is dying due to some force {it's believed to be dying in real life}, there's no way any of the races will escape in time because they cannot travel that fast, but then somehow the reapers aka immortal shepard helps them escape back to the milky way along with all the new intelligent species from andromeda) 

If you are referring to Lore wise then yes we still have Mass Effect fields and at the very least Biotics.

As to actual game data that would carry over from any ME1, ME2, ME3?
No.
There is no connection.
No ME Keep.

The DA Keep was used to carry choices into Inquisition from DA:O and DA2 -which ALL took place on the same stage just at different times into DA:I.

ME:A takes place in a future time and is also far off on a completely different stage.

This is my expectation.

Happened in the shuttle at the start of Mass Effect 2.
The Witcher III did pretty much this after the first Act.