When making knife blocks and similar sorts of glue-up situations there are multiple tricks that can minimise or prevent glue ending up in the milled grooves, and I think it's normal to use a combination rather than attempt to rely exclusively on one because just one can't be guaranteed to be effective.
First and foremost I think is simply: don't apply too much glue, i.e. not the normal amount one would use when laminating for example.
Normally in woodworking you most definitely want to apply at least a slight excess of glue1, but here you can try to apply just enough, sacrificing absolute strength for ease of assembly/minimal cleanup.
Glue as normal, then wipe the glue from around the grooves
I've used this basic trick many times, but unfortunately with varying levels of success. So although it sometimes works great IME it isn't a sure-fire way to prevent glue ending up where you don't want it; combine it with another tip or expect to have to clean up some visible glue in the grooves.
The normal tool used by woodworkers to wipe excess glue away seems to be a fingertip :-) but a wooden scraper or silicone-rubber spatula could work as well or better.
Wax or finish the grooves, then assemble twice
This will only work if the decorative sheet is thick enough to allow it to be separated without being damaged; obviously this trick can't be used if you plan to use anything like a veneer thickness.
Here you basically glue as normal, press the parts together, separate them to clean out the glue from the grooves (hence why you wax or apply finish to them beforehand), then bring the parts together one final time and apply full clamping pressure.
You may (in fact are likely to) get some minimal squeezeout along the edges of the grooves when full clamping pressure is applied — little drops dotted along the edge — but any you can't reach from the top to clean up will be A) invisible in use and B) very unlikely to be large enough to cause an issue given the very generous sizing of the slots.
Fill the grooves with something first
If you fill the grooves with close-fitting strips of plastic, or a fine-grained hardwood (maple would be ideal) well waxed to prevent glue sticking to them.
This might work well enough by itself that it could be used solo, assuming making the strips of plastic or wood isn't an insurmountable obstacle2.
Give somewhere else for the glue to go
This trick is not new, but seems to be gaining increasing traction in recent years as a method to minimise or eliminate squeezeout when certain joints are assembled.
If you want to employ this here I think the way to do it is to create very shallow stopped grooves alongside the knife grooves. The first method to do this that I thought of was to use a router (either hand or powered), but actually just cutting V-grooves using a sharp knife and a straightedge might be all that's needed, and this would be far faster and require no setup.
Use contact adhesive
I wanted to include this for completeness, although I wouldn't favour this method myself as although the initial bond will be plenty strong I'm not confident it would hold up over the years, plus the glue line might be thicker than you'd really like it to be, so visually intrusive.
But in its favour, because contact adhesive is applied to both faces and then allowed to dry, you essentially can't get squeezout. Of course you're likely to get some adhesive in the grooves during application, but that's relatively easy to prevent, or remove while the joint is still open.
Note applicable with any glue used
I think it's possible this might not be immediately obvious so do bear in mind you wouldn't want to apply glue to the entire of the decorative sheet, because you'd end up with glue on one side of every knife slot. This would be particularly unwanted in the case of contact adhesive since this remains tacky.
To prevent this you'll need to carefully mark the positions of the slots on the inside face of the sheet and either mask them off before spreading the glue, or brush around them.
1 Excess glue being mostly — or ideally all — squeezed out as the clamps are fully tightened. This is what gives the best chance of the strongest joint being formed with the two most common woodworking adhesives, yielding no dry spots (AKA 'starved' areas) which are weak points.
2 Impossible or too tedious to be bothered with :-)