Set up a search like:

~artistname ~artistname ~artistname order:id id:>0

Add in the artists you want to watch. You'll get all of their posts in the order they were posted, oldest to newest. Decide on a starting point (or just take the most recent one if you want to start from today), replace the 0 at the end with the ID of that post, and bookmark it.

When you want to check for new stuff, open the bookmark, and you'll see all posts by those artists that came after the one you took the ID from. Browse as you please, and when you get to the end, copy the ID of the most recent post, replace the ID in your search, and update the bookmark.

Also works for any search that isn't artist names, too, obviously.

I've been using the follow feature of e1547 for Android. I'll be sure to add this as a feature to the desktop e621 client I'm developing (stay tuned for that. I'll have it ready to publish any month now, just as soon as uni gives me a breather....)

I use the RE621 userscript. It can be found on the e621 forum under "Tools and Applications"

Bookmark them all. And then you sort by "most recent visit" so you see the artists you haven't checked out in a while

I use https://t.me/E621WatchBot on telegram to get notifications of new uploads matching queries. (You can specify an artist’s tag as a query.)