Even without a gameboard, and even without finely tracking movement at all, you can still have players with movement builds and make them feel powerful. Describing opponents like longbow users and warmages as being within range of a move action "for you, but not for the others" is just one basic way to make them feel fast. Things like difficult terrain being an effective nonissue for them is good to keep in mind. Just remember to describe how fast they are in combat, and give them fun things to do with that speed.
There are so many free(mium) alternatives for playing RPGs and specifically 5e over the internet that playing virtually without a map is almost a conscious choice*.
ROLL20
USE ROLL20
You don't need to take the game itself off Discord, but Roll20 is free and does exactly what you want it to do: Easily track the position of everything. It's what I do and it improved my experience to the point it felt like a whole different game.
This is how we play, video chat in discord, combat and exploration in roll 20
I've played in a DnD group using Discord before. Google Sheets works surprisingly well as a gameboard. It also allowed us to move characters and point stuff out remotely, since it supports simultaneous live editing.
That sounds creative, yet horrifying
You've gotten some good advice, but I want to add that if you're feeling masochistic you can use the discord bot Avrae to do your maps in discord chat.
Joking aside, Avrae is great and I really recommend you check it out if you aren't using it... but using it for maps is kinda miserable. Using a dedicated virtual tabletop like Owlbear Rodeo will serve you much better.
Owlbear. Rodeo.
I open a map on paint.net, overlay a grid on it if it hasn't one already, and place player/enemy tokens on it. Then I stream it to all the group.
It's 100% free and it's relatively quick to set up and run.
What anime is this
I think this has to be Attack on Titan, right?
That would be correct
In case anyone else wanted to use this meme template, I found this: https://imgflip.com/memegenerator/365536424/Yelena-disgust-face
