It's already established in the lore that you don't *need* relays for FTL travel. It's just they they're faster and much more efficient at long distance because ship's FTL'ing it in open space are limited by their need to periodically discharge their engines before they literally cook the crew alive.
Indeed, we saw that between Arrival and ME3 that the Reapers weren't restricted by being denied access to the relays (first the Citadel itself and later the Alpha relay) it only slowed them down.
Also worth remembering that half the point of the relay network was specifically so that the emergent space faring races wouldn't bother exploring other avenues of FTL travel, making the harvest cycles more efficient and technological progress more predicable.
Whatever colour you picked at the end of ME3, there's going to be plenty of reaper tech and refined eezo lying around for the milky-way races to reverse engineer and produce their own trans-galactic engines.
That's not to say that there can't be relays out there. Indeed, if this really is about an effort to explore and colonise Andromeda then part of the plot could be the building of a new network that links the galaxies together.
You'd only need one long range ship to make the crossing (even if it takes centuries, that's not much of a problem if it's crewed by Asari & Krogan) and build the first relay. Once you have that beachhead other ships can start pouring through and spreading out system by system.
ETA: On the off chance anyone is an any doubt as to the time frame, there's
this article just posted on the Bioware Blog. (emphasis mine)
While we aren’t ready to go into too many details just yet, as you saw in the trailer and can tell by the name, this game is very much a new adventure, taking place far away from and long after the events of the original trilogy. You will play a human, male or female, though that’s actually not the character you saw in the trailer (more on that later). You’ll be exploring an all-new galaxy, Andromeda, and piloting the new and improved Mako you saw. And through it all, you will have a new team of adventurers to work with, learn from, fight alongside of, and fall in love with.
I'm not too crazy about the new PC being strictly human. I had hoped at least Asari and Turian might also be options.
Still, it's interesting that they point out that the character we saw isn't actually the PC. Could Shepard be alive after all, at least in some form? Aside from the N7 marking, that's pretty much
the same pose and shot they used for the first ME3 teaser. Honestly I doubt it, but still....