Look at which browsers your users are actually using, and check https://caniuse.com for which versions of which browsers support the new feature. Make a business decision on what percentage of visitors you're okay with getting a page without a nice layout.
If the number of visitors with non-supporting browsers is tiny, then go for it! But make sure at the very least your pages fall back to something where the content is visible and accessible... like make sure it is at least somewhat usable as a single-column list of things.
If the number of visitors with non-supporting browsers is higher than you're willing to lose out on, then still use CSS grid but spend more time on a fallback that looks okay (but doesn't have to be perfect). Perhaps this could be using flexbox (if your users are mostly on kind-of-new-but-not-super-old browsers), otherwise go old-school and use either floats or display:table. (I personally would go with display:table since it's the most grid-like... but depending on your layout it might fall apart under certain circumstances).
Key things to keep in mind are:
1) There are still plenty of people using browsers that don't support CSS Grid. But this may or may not be plenty of YOUR users, so check your actual traffic.
2) You definitely want a fallback of some kind, but the fallback does NOT need to be exactly perfect! "Good enough" is probably good enough.
And hopefully in 3 or 4 years we can forget about a time when CSS didn't actually have any tools intended for page layout :)
The evergreen browsers (chrome, ff, edge, safari, etc...) were well supported.
This of it this way: If your mobile view is also your <IE11 view, you shouldn't have to do much overriding.
You will need fallbacks. Flexbox isn't going anywhere.