I have several 22 rifles, including two Marlin bolt actions. One is a relatively new XT22M (22 WMR) and the other is an older 25N (22LR). While both are perfectly serviceable rifles, I'm not sure the microgroove rifling is any specific advantage. The XT22M is the least accurate of my three 22 mag rifles, albeit not by much. It does OK at 100 yards however, doing 1.3-1.6" groups with CCI or Winchester 40g ammo; CCI and Hornady 30g ammo doesn't do as well, averaging right around 2" groups at that distance. It is limited by a 3-9 scope and me shooting it.
The older 25N is in the middle of the pack among my 22 LR bolt guns. It averages 1" groups at 50 yards with a 2.5-7 scope, using bulk ammo, e.g., CCI SV, Federal AutoMatch, or Winchester M22.
However, they were both relatively inexpensive (about $110 for the 25N when new and $190 for the XT22M a couple years ago. My least expensive bolt guns. I did replace the trigger spring in the old 25N to lower the pull weight, and it shoots better now. The current XT series has the newer and much lighter blade trigger on it. Much nicer to shoot than the old 25 series triggers.
Will a Marlin bolt action be more accurate than what you have now? Quite frankly, it all depends on the specific rifle you get. It may be astounding, as Signalman's example above. More likely, it will be adequate but not stellar. But it will be fun and won't cost a lot.
The next step up, in my opinion, would be a Ruger American Rimfire. I have a Compact model in 22LR and a Standard model in 22WMR. Both were $260 when they first came out. The 22WMR shot great right out of the box, which convinced me I needed one in 22LR. Unfortunately, my specific 22LR Compact model didn't shoot quite as well, although it is improving. Right now it shoots just about the same as my old Marlin 25N. But the RARs are loaded with nice features that the Marlins don't have, like the bedding blocks, interchangeable stock modules, a solid feeling stock (for an inexpensive synthetic stock), and nice trigger. And those handy Ruger rotary magazines.